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<span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Types</span> </div> </a> <button aria-controls="toc-Types-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types subsection</span> </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Ordinary_differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordinary_differential_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Ordinary differential equations</span> </div> </a> <ul id="toc-Ordinary_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partial_differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_differential_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Partial differential equations</span> </div> </a> <ul id="toc-Partial_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-linear_differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-linear_differential_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Non-linear differential equations</span> </div> </a> <ul id="toc-Non-linear_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equation_order_and_degree" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equation_order_and_degree"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Equation order and degree</span> </div> </a> <ul id="toc-Equation_order_and_degree-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Existence_of_solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Existence_of_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Existence of solutions</span> </div> </a> <ul id="toc-Existence_of_solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Related concepts</span> </div> </a> <ul id="toc-Related_concepts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_to_difference_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Connection_to_difference_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Connection to difference equations</span> </div> </a> <ul id="toc-Connection_to_difference_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Software" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Software"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Software</span> </div> </a> <ul id="toc-Software-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Differential equation</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 95 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-95" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">95 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Differensiaalvergelyking" title="Differensiaalvergelyking – Afrikaans" lang="af" hreflang="af" data-title="Differensiaalvergelyking" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Differentialgleichung" title="Differentialgleichung – Alemannic" lang="gsw" hreflang="gsw" data-title="Differentialgleichung" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%AA%D9%81%D8%A7%D8%B6%D9%84%D9%8A%D8%A9" title="معادلة تفاضلية – Arabic" lang="ar" hreflang="ar" data-title="معادلة تفاضلية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Ecuaci%C3%B3n_diferencial" title="Ecuación diferencial – Aragonese" lang="an" hreflang="an" data-title="Ecuación diferencial" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%8F%D5%A1%D6%80%D5%A2%D5%A5%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B0%D5%A1%D6%82%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Տարբերական հաւասարումներ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Տարբերական հաւասարումներ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Ecuaci%C3%B3n_diferencial" title="Ecuación diferencial – Asturian" lang="ast" hreflang="ast" data-title="Ecuación diferencial" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Diferensial_t%C9%99nlikl%C9%99r" title="Diferensial tənliklər – Azerbaijani" lang="az" hreflang="az" data-title="Diferensial tənliklər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%AC%E0%A6%95%E0%A6%B2%E0%A6%A8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="ব্যবকলনীয় সমীকরণ – Bangla" lang="bn" hreflang="bn" data-title="ব্যবকলনীয় সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/B%C3%AE-hun_hong-t%C3%AAng-sek" title="Bî-hun hong-têng-sek – Minnan" lang="nan" hreflang="nan" data-title="Bî-hun hong-têng-sek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C_%D1%82%D0%B8%D0%B3%D0%B5%D2%99%D0%BB%D3%99%D0%BC%D3%99" title="Дифференциаль тигеҙләмә – Bashkir" lang="ba" hreflang="ba" data-title="Дифференциаль тигеҙләмә" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8B%D1%84%D0%B5%D1%80%D1%8D%D0%BD%D1%86%D1%8B%D1%8F%D0%BB%D1%8C%D0%BD%D0%B0%D0%B5_%D1%9E%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D0%B5" title="Дыферэнцыяльнае ўраўненне – Belarusian" lang="be" hreflang="be" data-title="Дыферэнцыяльнае ўраўненне" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%94%D1%8B%D1%84%D1%8D%D1%80%D1%8D%D0%BD%D1%86%D1%8B%D0%B9%D0%BD%D0%B0%D0%B5_%D1%80%D0%B0%D1%9E%D0%BD%D0%B0%D0%BD%D1%8C%D0%BD%D0%B5" title="Дыфэрэнцыйнае раўнаньне – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дыфэрэнцыйнае раўнаньне" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%BE_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Диференциално уравнение – Bulgarian" lang="bg" hreflang="bg" data-title="Диференциално уравнение" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Diferencijalna_jedna%C4%8Dina" title="Diferencijalna jednačina – Bosnian" lang="bs" hreflang="bs" data-title="Diferencijalna jednačina" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_diferencial" title="Equació diferencial – Catalan" lang="ca" hreflang="ca" data-title="Equació diferencial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BB%C4%83_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85" title="Дифференциаллă танлăх – Chuvash" lang="cv" hreflang="cv" data-title="Дифференциаллă танлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diferenci%C3%A1ln%C3%AD_rovnice" title="Diferenciální rovnice – Czech" lang="cs" hreflang="cs" data-title="Diferenciální rovnice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Hafaliad_differol" title="Hafaliad differol – Welsh" lang="cy" hreflang="cy" data-title="Hafaliad differol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Differentialligning" title="Differentialligning – Danish" lang="da" hreflang="da" data-title="Differentialligning" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Differentialgleichung" title="Differentialgleichung – German" lang="de" hreflang="de" data-title="Differentialgleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Diferentsiaalv%C3%B5rrand" title="Diferentsiaalvõrrand – Estonian" lang="et" hreflang="et" data-title="Diferentsiaalvõrrand" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CF%86%CE%BF%CF%81%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7" title="Διαφορική εξίσωση – Greek" lang="el" hreflang="el" data-title="Διαφορική εξίσωση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_diferencial" title="Ecuación diferencial – Spanish" lang="es" hreflang="es" data-title="Ecuación diferencial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diferenciala_ekvacio" title="Diferenciala ekvacio – Esperanto" lang="eo" hreflang="eo" data-title="Diferenciala ekvacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ekuazio_diferentzial" title="Ekuazio diferentzial – Basque" lang="eu" hreflang="eu" data-title="Ekuazio diferentzial" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%AF%DB%8C%D9%81%D8%B1%D8%A7%D9%86%D8%B3%DB%8C%D9%84" title="معادله دیفرانسیل – Persian" lang="fa" hreflang="fa" data-title="معادله دیفرانسیل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Differential_equation" title="Differential equation – Fiji Hindi" lang="hif" hreflang="hif" data-title="Differential equation" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_diff%C3%A9rentielle" title="Équation différentielle – French" lang="fr" hreflang="fr" data-title="Équation différentielle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cothrom%C3%B3id_dhifre%C3%A1lach" title="Cothromóid dhifreálach – Irish" lang="ga" hreflang="ga" data-title="Cothromóid dhifreálach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3n_diferencial" title="Ecuación diferencial – Galician" lang="gl" hreflang="gl" data-title="Ecuación diferencial" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程 – Gan" lang="gan" hreflang="gan" data-title="微分方程" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AF%B8%EB%B6%84%EB%B0%A9%EC%A0%95%EC%8B%9D" title="미분방정식 – Korean" lang="ko" hreflang="ko" data-title="미분방정식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D6%86%D5%A5%D6%80%D5%A5%D5%B6%D6%81%D5%AB%D5%A1%D5%AC_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Դիֆերենցիալ հավասարումներ – Armenian" lang="hy" hreflang="hy" data-title="Դիֆերենցիալ հավասարումներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="अवकल समीकरण – Hindi" lang="hi" hreflang="hi" data-title="अवकल समीकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Diferencijalne_jednad%C5%BEbe" title="Diferencijalne jednadžbe – Croatian" lang="hr" hreflang="hr" data-title="Diferencijalne jednadžbe" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_diferensial" title="Persamaan diferensial – Indonesian" lang="id" hreflang="id" data-title="Persamaan diferensial" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Diffurjafna" title="Diffurjafna – Icelandic" lang="is" hreflang="is" data-title="Diffurjafna" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_differenziale" title="Equazione differenziale – Italian" lang="it" hreflang="it" data-title="Equazione differenziale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="משוואה דיפרנציאלית – Hebrew" lang="he" hreflang="he" data-title="משוואה דיפרנציאלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%A4%E1%83%94%E1%83%A0%E1%83%94%E1%83%9C%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%92%E1%83%90%E1%83%9C%E1%83%A2%E1%83%9D%E1%83%9A%E1%83%94%E1%83%91%E1%83%94%E1%83%91%E1%83%98" title="დიფერენციალური განტოლებები – Georgian" lang="ka" hreflang="ka" data-title="დიფერენციალური განტოლებები" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%B4%D1%8B%D2%9B_%D1%82%D0%B5%D2%A3%D0%B4%D0%B5%D1%83" title="Дифференциалдық теңдеу – Kazakh" lang="kk" hreflang="kk" data-title="Дифференциалдық теңдеу" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mlinganyo_tenguo" title="Mlinganyo tenguo – Swahili" lang="sw" hreflang="sw" data-title="Mlinganyo tenguo" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hevk%C3%AA%C5%9Feya_d%C3%AEferensiyel" title="Hevkêşeya dîferensiyel – Kurdish" lang="ku" hreflang="ku" data-title="Hevkêşeya dîferensiyel" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Aequatio_differentialis" title="Aequatio differentialis – Latin" lang="la" hreflang="la" data-title="Aequatio differentialis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Diferenci%C4%81lvien%C4%81dojums" title="Diferenciālvienādojums – Latvian" lang="lv" hreflang="lv" data-title="Diferenciālvienādojums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB_%D0%B1%D0%B0%D1%80%D0%B0%D0%B1%D0%B0%D1%80%D0%B2%D0%B0%D0%BB" title="Дифференциал барабарвал – Lezghian" lang="lez" hreflang="lez" data-title="Дифференциал барабарвал" data-language-autonym="Лезги" data-language-local-name="Lezghian" class="interlanguage-link-target"><span>Лезги</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Diferencialin%C4%97_lygtis" title="Diferencialinė lygtis – Lithuanian" lang="lt" hreflang="lt" data-title="Diferencialinė lygtis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Equazzion_diferenziala" title="Equazzion diferenziala – Lombard" lang="lmo" hreflang="lmo" data-title="Equazzion diferenziala" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Differenci%C3%A1legyenlet" title="Differenciálegyenlet – Hungarian" lang="hu" hreflang="hu" data-title="Differenciálegyenlet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B0" title="Диференцијална равенка – Macedonian" lang="mk" hreflang="mk" data-title="Диференцијална равенка" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%B5%E0%B4%95%E0%B4%B2%E0%B4%B8%E0%B4%AE%E0%B4%B5%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%AF%E0%B4%82" title="അവകലസമവാക്യം – Malayalam" lang="ml" hreflang="ml" data-title="അവകലസമവാക്യം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Ekwazzjoni_differenzjali" title="Ekwazzjoni differenzjali – Maltese" lang="mt" hreflang="mt" data-title="Ekwazzjoni differenzjali" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Persamaan_pembezaan" title="Persamaan pembezaan – Malay" lang="ms" hreflang="ms" data-title="Persamaan pembezaan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Differentiaalvergelijking" title="Differentiaalvergelijking – Dutch" lang="nl" hreflang="nl" data-title="Differentiaalvergelijking" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%BC%8F" title="微分方程式 – Japanese" lang="ja" hreflang="ja" data-title="微分方程式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Differensialligning" title="Differensialligning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Differensialligning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Differensiallikning" title="Differensiallikning – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Differensiallikning" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Equacion_diferenciala" title="Equacion diferenciala – Occitan" lang="oc" hreflang="oc" data-title="Equacion diferenciala" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Differensial_tenglama" title="Differensial tenglama – Uzbek" lang="uz" hreflang="uz" data-title="Differensial tenglama" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A1%E0%A8%BF%E0%A8%AB%E0%A8%BC%E0%A8%B0%E0%A9%88%E0%A8%82%E0%A8%B8%E0%A8%BC%E0%A9%80%E0%A8%85%E0%A8%B2_%E0%A8%B8%E0%A8%AE%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8" title="ਡਿਫ਼ਰੈਂਸ਼ੀਅਲ ਸਮੀਕਰਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਡਿਫ਼ਰੈਂਸ਼ੀਅਲ ਸਮੀਕਰਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AA%D9%81%D8%B1%DB%8C%D9%82%DB%8C_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA" title="تفریقی مساوات – Western Punjabi" lang="pnb" hreflang="pnb" data-title="تفریقی مساوات" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Difrenshal_ikwiejan" title="Difrenshal ikwiejan – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Difrenshal ikwiejan" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9F%E1%9E%98%E1%9E%B8%E1%9E%80%E1%9E%B6%E1%9E%9A%E1%9E%8C%E1%9E%B8%E1%9E%95%E1%9F%81%E1%9E%9A%E1%9F%89%E1%9E%84%E1%9F%8B%E1%9E%9F%E1%9F%92%E1%9E%99%E1%9F%82%E1%9E%9B" title="សមីការឌីផេរ៉ង់ស្យែល – Khmer" lang="km" hreflang="km" data-title="សមីការឌីផេរ៉ង់ស្យែល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Equassion_diferensial" title="Equassion diferensial – Piedmontese" lang="pms" hreflang="pms" data-title="Equassion diferensial" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_r%C3%B3%C5%BCniczkowe" title="Równanie różniczkowe – Polish" lang="pl" hreflang="pl" data-title="Równanie różniczkowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%A3o_diferencial" title="Equação diferencial – Portuguese" lang="pt" hreflang="pt" data-title="Equação diferencial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Differensial_te%C5%84leme" title="Differensial teńleme – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Differensial teńleme" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Bie_diferen%C8%9Bial%C4%83" title="Ecuație diferențială – Romanian" lang="ro" hreflang="ro" data-title="Ecuație diferențială" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Дифференциальное уравнение – Russian" lang="ru" hreflang="ru" data-title="Дифференциальное уравнение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Differential_equation" title="Differential equation – Scots" lang="sco" hreflang="sco" data-title="Differential equation" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ekuacionet_diferenciale" title="Ekuacionet diferenciale – Albanian" lang="sq" hreflang="sq" data-title="Ekuacionet diferenciale" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Equazzioni_diffirinziali" title="Equazzioni diffirinziali – Sicilian" lang="scn" hreflang="scn" data-title="Equazzioni diffirinziali" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%85%E0%B7%80%E0%B6%9A%E0%B6%BD_%E0%B7%83%E0%B6%B8%E0%B7%93%E0%B6%9A%E0%B6%BB%E0%B6%AB%E0%B6%BA" title="අවකල සමීකරණය – Sinhala" lang="si" hreflang="si" data-title="අවකල සමීකරණය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Differential_equation" title="Differential equation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Differential equation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Diferenci%C3%A1lna_rovnica" title="Diferenciálna rovnica – Slovak" lang="sk" hreflang="sk" data-title="Diferenciálna rovnica" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Diferencialna_ena%C4%8Dba" title="Diferencialna enačba – Slovenian" lang="sl" hreflang="sl" data-title="Diferencialna enačba" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%DA%A9%DB%8E%D8%B4%DB%95%DB%8C_%D8%AC%DB%8C%D8%A7%DA%A9%D8%A7%D8%B1%DB%8C" title="ھاوکێشەی جیاکاری – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ھاوکێشەی جیاکاری" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B0" title="Диференцијална једначина – Serbian" lang="sr" hreflang="sr" data-title="Диференцијална једначина" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Diferencijalna_jedna%C4%8Dina" title="Diferencijalna jednačina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Diferencijalna jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / 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class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Diferansiyel_denklem" title="Diferansiyel denklem – Turkish" lang="tr" hreflang="tr" data-title="Diferansiyel denklem" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Differensial_de%C5%88lemeler" title="Differensial deňlemeler – Turkmen" lang="tk" hreflang="tk" data-title="Differensial deňlemeler" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F" title="Диференціальні рівняння – Ukrainian" lang="uk" hreflang="uk" data-title="Диференціальні рівняння" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D9%81%D8%B1%D9%82%DB%8C_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA" title="تفرقی مساوات – Urdu" lang="ur" hreflang="ur" data-title="تفرقی مساوات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Differencialine_tazostuz" title="Differencialine tazostuz – Veps" lang="vep" hreflang="vep" data-title="Differencialine tazostuz" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_vi_ph%C3%A2n" title="Phương trình vi phân – Vietnamese" lang="vi" hreflang="vi" data-title="Phương trình vi phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Ekwasyon_diferensyal" title="Ekwasyon diferensyal – Waray" lang="war" hreflang="war" data-title="Ekwasyon diferensyal" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程 – Wu" lang="wuu" hreflang="wuu" data-title="微分方程" data-language-autonym="吴语" data-language-local-name="Wu" 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searchaux" style="display:none">Type of functional equation (mathematics)</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference equation</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a class="mw-selflink selflink">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Scope</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;"> <div class="hlist"><ul><li><a href="/wiki/Natural_science" title="Natural science">Natural sciences</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></li> <li><a href="/wiki/Physics" title="Physics">Physics</a></li> <li><a href="/wiki/Chemistry" title="Chemistry">Chemistry</a></li> <li><br /><a href="/wiki/Biology" title="Biology">Biology</a></li> <li><a href="/wiki/Geology" title="Geology">Geology</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;"> <ul><li><a href="/wiki/Economics" title="Economics">Economics</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td> </tr></tbody></table> <hr /> <a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;"> Classification</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> By variable type</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li> <li>Coupled / Decoupled</li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a> / <a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> Features</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Operator</a></li></ul> </div> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference <span style="font-size:85%;">(discrete analogue)</span></a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Solution</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem </a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li><a href="/wiki/Initial_condition" title="Initial condition">Initial conditions</a></li> <li><a href="/wiki/Boundary_value_problem" title="Boundary value problem">Boundary values</a> <ul><li><a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li> <li><a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li> <li><a href="/wiki/Robin_boundary_condition" title="Robin boundary condition">Robin</a></li> <li><a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov</a> / <a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a> / <a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><span class="nowrap"><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a> / Integral solutions</span></li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li>Inspection</li> <li><a href="/wiki/Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li> <li><br /><a href="/wiki/Euler_method" title="Euler method">Euler</a></li> <li><a href="/wiki/Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference</a> <span style="font-size:85%;">(<a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</span></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element</a> <ul><li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element</a></li></ul></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin</a> <ul><li><a href="/wiki/Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin</a></li></ul></li> <li><a href="/wiki/Green%27s_function" title="Green's function">Green's function</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a></li></ul> </div> <ul><li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> People</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">List</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist" style="padding-top:0.5em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/George_Green_(mathematician)" title="George Green (mathematician)">George Green</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations" title="Template:Differential equations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations" title="Template talk:Differential equations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations" title="Special:EditPage/Template:Differential equations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>differential equation</b> is an <a href="/wiki/Functional_equation" title="Functional equation">equation</a> that relates one or more unknown <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> and their <a href="/wiki/Derivative" title="Derivative">derivatives</a>.<sup id="cite_ref-Zill2012_1-0" class="reference"><a href="#cite_note-Zill2012-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Economics" title="Economics">economics</a>, and <a href="/wiki/Biology" title="Biology">biology</a>. </p><p>The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. </p><p>Often when a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a> for the solutions is not available, solutions may be approximated numerically using computers. The <a href="/wiki/Dynamical_systems_theory" title="Dynamical systems theory">theory of dynamical systems</a> puts emphasis on <a href="https://en.wiktionary.org/wiki/qualitative" class="extiw" title="wikt:qualitative">qualitative</a> analysis of systems described by differential equations, while many <a href="/wiki/Numerical_methods" class="mw-redirect" title="Numerical methods">numerical methods</a> have been developed to determine solutions with a given degree of accuracy. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Differential equations came into existence with the <a href="/wiki/History_of_calculus" title="History of calculus">invention of calculus</a> by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a>. In Chapter 2 of his 1671 work <a href="/wiki/Method_of_Fluxions" title="Method of Fluxions"><i>Methodus fluxionum et Serierum Infinitarum</i></a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Newton listed three kinds of differential equations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=f(x)\\[4pt]{\frac {dy}{dx}}&=f(x,y)\\[4pt]x_{1}{\frac {\partial y}{\partial x_{1}}}&+x_{2}{\frac {\partial y}{\partial x_{2}}}=y\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=f(x)\\[4pt]{\frac {dy}{dx}}&=f(x,y)\\[4pt]x_{1}{\frac {\partial y}{\partial x_{1}}}&+x_{2}{\frac {\partial y}{\partial x_{2}}}=y\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17bdcadcae4f4602463dba2e7ad439b31c3fa5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:21.69ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=f(x)\\[4pt]{\frac {dy}{dx}}&=f(x,y)\\[4pt]x_{1}{\frac {\partial y}{\partial x_{1}}}&+x_{2}{\frac {\partial y}{\partial x_{2}}}=y\end{aligned}}}"></span></dd></dl> <p>In all these cases, <span class="texhtml mvar" style="font-style:italic;">y</span> is an unknown function of <span class="texhtml mvar" style="font-style:italic;">x</span> (or of <span class="texhtml"><i>x</i><sub>1</sub></span> and <span class="texhtml"><i>x</i><sub>2</sub></span>), and <span class="texhtml mvar" style="font-style:italic;">f</span> is a given function. </p><p>He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. </p><p><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a> proposed the <a href="/wiki/Bernoulli_differential_equation" title="Bernoulli differential equation">Bernoulli differential equation</a> in 1695.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This is an <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'+P(x)y=Q(x)y^{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'+P(x)y=Q(x)y^{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df4fcdfdb40fe6b609da7321a8c1d1a63c90eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.567ex; height:3.009ex;" alt="{\displaystyle y'+P(x)y=Q(x)y^{n}\,}"></span></dd></dl> <p>for which the following year Leibniz obtained solutions by simplifying it.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Historically, the problem of a vibrating string such as that of a <a href="/wiki/Musical_instrument" title="Musical instrument">musical instrument</a> was studied by <a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a>, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>, and <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In 1746, d’Alembert discovered the one-dimensional <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>, and within ten years Euler discovered the three-dimensional wave equation.<sup id="cite_ref-Speiser_9-0" class="reference"><a href="#cite_note-Speiser-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</a> was developed in the 1750s by Euler and Lagrange in connection with their studies of the <a href="/wiki/Tautochrone" class="mw-redirect" title="Tautochrone">tautochrone</a> problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, which led to the formulation of <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>. </p><p>In 1822, <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier</a> published his work on <a href="/wiki/Heat_flow" class="mw-redirect" title="Heat flow">heat flow</a> in <i>Théorie analytique de la chaleur</i> (The Analytic Theory of Heat),<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in which he based his reasoning on <a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</a>, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a> for conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum. </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the motion of a body is described by its position and velocity as the time value varies. <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws</a> allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time. </p><p>In some cases, this differential equation (called an <a href="/wiki/Equations_of_motion" title="Equations of motion">equation of motion</a>) may be solved explicitly. </p><p>An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity. </p> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=3" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts. </p> <div class="mw-heading mw-heading3"><h3 id="Ordinary_differential_equations">Ordinary differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=4" title="Edit section: Ordinary differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a> and <a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear differential equation</a></div> <p>An <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> (<i>ODE</i>) is an equation containing an unknown <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">function of one real or complex variable</a> <span class="texhtml mvar" style="font-style:italic;">x</span>, its derivatives, and some given functions of <span class="texhtml mvar" style="font-style:italic;">x</span>. The unknown function is generally represented by a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a> (often denoted <span class="texhtml mvar" style="font-style:italic;">y</span>), which, therefore, <i>depends</i> on <span class="texhtml mvar" style="font-style:italic;">x</span>. Thus <span class="texhtml mvar" style="font-style:italic;">x</span> is often called the <a href="/wiki/Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a> of the equation. The term "<i>ordinary</i>" is used in contrast with the term <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a>, which may be with respect to <i>more than</i> one independent variable. </p><p><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear differential equations</a> are the differential equations that are <a href="/wiki/Linear_equation" title="Linear equation">linear</a> in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of <a href="/wiki/Antiderivative" title="Antiderivative">integrals</a>. </p><p>Most ODEs that are encountered in <a href="/wiki/Physics" title="Physics">physics</a> are linear. Therefore, most <a href="/wiki/Special_functions" title="Special functions">special functions</a> may be defined as solutions of linear differential equations (see <a href="/wiki/Holonomic_function" title="Holonomic function">Holonomic function</a>). </p><p>As, in general, the solutions of a differential equation cannot be expressed by a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a>, <a href="/wiki/Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">numerical methods</a> are commonly used for solving differential equations on a computer. </p> <div class="mw-heading mw-heading3"><h3 id="Partial_differential_equations">Partial differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=5" title="Edit section: Partial differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></div> <p>A <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a> (<i>PDE</i>) is a differential equation that contains unknown <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable functions</a> and their <a href="/wiki/Partial_derivatives" class="mw-redirect" title="Partial derivatives">partial derivatives</a>. (This is in contrast to <a href="/wiki/Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a>, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant <a href="/wiki/Computer_model" class="mw-redirect" title="Computer model">computer model</a>. </p><p>PDEs can be used to describe a wide variety of phenomena in nature such as <a href="/wiki/Sound" title="Sound">sound</a>, <a href="/wiki/Heat" title="Heat">heat</a>, <a href="/wiki/Electrostatics" title="Electrostatics">electrostatics</a>, <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, <a href="/wiki/Fluid_flow" class="mw-redirect" title="Fluid flow">fluid flow</a>, <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elasticity</a>, or <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>, partial differential equations often model <a href="/wiki/Multidimensional_systems" class="mw-redirect" title="Multidimensional systems">multidimensional systems</a>. <a href="/wiki/Stochastic_partial_differential_equations" class="mw-redirect" title="Stochastic partial differential equations">Stochastic partial differential equations</a> generalize partial differential equations for modeling <a href="/wiki/Randomness" title="Randomness">randomness</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Non-linear_differential_equations">Non-linear differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=6" title="Edit section: Non-linear differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Non-linear_differential_equations" class="mw-redirect" title="Non-linear differential equations">Non-linear differential equations</a></div> <p>A <b>non-linear differential equation</b> is a differential equation that is not a <a href="/wiki/Linear_equation" title="Linear equation">linear equation</a> in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular <a href="/wiki/Symmetry" title="Symmetry">symmetries</a>. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of <a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a>. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. <a href="/wiki/Navier%E2%80%93Stokes_existence_and_smoothness" title="Navier–Stokes existence and smoothness">Navier–Stokes existence and smoothness</a>). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Linear differential equations frequently appear as <a href="/wiki/Linearization" title="Linearization">approximations</a> to nonlinear equations. These approximations are only valid under restricted conditions. For example, the <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations. </p> <div class="mw-heading mw-heading3"><h3 id="Equation_order_and_degree">Equation order and degree<span class="anchor" id="Second_order"></span><span class="anchor" id="Order"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=7" title="Edit section: Equation order and degree"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>order of the differential equation</b> is the highest <i><a href="/wiki/Order_of_derivative" class="mw-redirect" title="Order of derivative">order of derivative</a></i> of the unknown function that appears in the differential equation. For example, an equation containing only <a href="/wiki/First-order_derivative" class="mw-redirect" title="First-order derivative">first-order derivatives</a> is a <i><a href="/wiki/First-order_differential_equation" class="mw-redirect" title="First-order differential equation">first-order differential equation</a></i>, an equation containing the <a href="/wiki/Second-order_derivative" class="mw-redirect" title="Second-order derivative">second-order derivative</a> is a <i>second-order differential equation</i>, and so on.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>When it is written as a <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a> in the unknown function and its derivatives, its <b>degree of the differential equation</b> is, depending on the context, the <a href="/wiki/Polynomial_degree" class="mw-redirect" title="Polynomial degree">polynomial degree</a> in the highest derivative of the unknown function,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> or its <a href="/wiki/Total_degree" class="mw-redirect" title="Total degree">total degree</a> in the unknown function and its derivatives. In particular, a <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equation</a> has degree one for both meanings, but the non-linear differential equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'+y^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'+y^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2fe8e10eae179618276c80af8b6d4d60efe79b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.161ex; height:3.009ex;" alt="{\displaystyle y'+y^{2}=0}"></span> is of degree one for the first meaning but not for the second one. </p><p>Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the <a href="/wiki/Thin-film_equation" title="Thin-film equation">thin-film equation</a>, which is a fourth order partial differential equation. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=8" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the first group of examples <i>u</i> is an unknown function of <i>x</i>, and <i>c</i> and <i>ω</i> are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between <i><a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear</a></i> and <i>nonlinear</i> differential equations, and between <a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation"><i>homogeneous</i> differential equations</a> and <i>heterogeneous</i> ones. </p> <ul><li>Heterogeneous first-order linear constant coefficient ordinary differential equation: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {du}{dx}}=cu+x^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>c</mi> <mi>u</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {du}{dx}}=cu+x^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945804f346250140666c0a6523fd20eedf499eb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.688ex; height:5.509ex;" alt="{\displaystyle {\frac {du}{dx}}=cu+x^{2}.}"></span></dd></dl></li> <li>Homogeneous second-order linear ordinary differential equation: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}u}{dx^{2}}}-x{\frac {du}{dx}}+u=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>u</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}u}{dx^{2}}}-x{\frac {du}{dx}}+u=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173b1430314cc300bf90b4a967eae6380711b47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.068ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}u}{dx^{2}}}-x{\frac {du}{dx}}+u=0.}"></span></dd></dl></li> <li>Homogeneous second-order linear constant coefficient ordinary differential equation describing the <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}u}{dx^{2}}}+\omega ^{2}u=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}u}{dx^{2}}}+\omega ^{2}u=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7cdb026da11e5a24ca46f43bb8af325e5f3289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.016ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}u}{dx^{2}}}+\omega ^{2}u=0.}"></span></dd></dl></li> <li>Heterogeneous first-order nonlinear ordinary differential equation: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {du}{dx}}=u^{2}+4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {du}{dx}}=u^{2}+4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d03496c5764c42adfecd1c9660d876a4d0670e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.514ex; height:5.509ex;" alt="{\displaystyle {\frac {du}{dx}}=u^{2}+4.}"></span></dd></dl></li> <li>Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a <a href="/wiki/Pendulum" title="Pendulum">pendulum</a> of length <i>L</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L{\frac {d^{2}u}{dx^{2}}}+g\sin u=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mi>sin</mi> <mo>⁡</mo> <mi>u</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L{\frac {d^{2}u}{dx^{2}}}+g\sin u=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3272632b022f8ed937acc24597fb8dc386faf388" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.845ex; height:6.009ex;" alt="{\displaystyle L{\frac {d^{2}u}{dx^{2}}}+g\sin u=0.}"></span></dd></dl></li></ul> <p>In the next group of examples, the unknown function <i>u</i> depends on two variables <i>x</i> and <i>t</i> or <i>x</i> and <i>y</i>. </p> <ul><li>Homogeneous first-order linear partial differential equation: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial t}}+t{\frac {\partial u}{\partial x}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial t}}+t{\frac {\partial u}{\partial x}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf368fadab981484268ec32723105c65a1e7028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.556ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial u}{\partial t}}+t{\frac {\partial u}{\partial x}}=0.}"></span></dd></dl></li> <li>Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the <a href="/wiki/Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de38a82a39e63e77a24a423a9e430924b3539b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.875ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.}"></span></dd></dl></li> <li>Homogeneous third-order non-linear partial differential equation, the <a href="/wiki/Korteweg%E2%80%93De_Vries_equation" title="Korteweg–De Vries equation">KdV equation</a>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial t}}=6u{\frac {\partial u}{\partial x}}-{\frac {\partial ^{3}u}{\partial x^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>6</mn> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial t}}=6u{\frac {\partial u}{\partial x}}-{\frac {\partial ^{3}u}{\partial x^{3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe49befc15be8cc358653332fde7f46398699ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.609ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial t}}=6u{\frac {\partial u}{\partial x}}-{\frac {\partial ^{3}u}{\partial x^{3}}}.}"></span></dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="Existence_of_solutions">Existence of solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=9" title="Edit section: Existence of solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Equation_solving" title="Equation solving">Solving</a> differential equations is not like solving <a href="/wiki/Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a>. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest. </p><p>For first order initial value problems, the <a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a> gives one set of circumstances in which a solution exists. Given any point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> in the xy-plane, define some rectangular region <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>, such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=[l,m]\times [n,p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>l</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=[l,m]\times [n,p]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/515b71ad3b5fc01ff532df8cb71018baca811973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.572ex; height:2.843ex;" alt="{\displaystyle Z=[l,m]\times [n,p]}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"></span> is in the interior of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>. If we are given a differential equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {dy}{dx}}=g(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {dy}{dx}}=g(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55300534dfeb4b0220920df1cc3eacba0764f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.179ex; height:4.176ex;" alt="{\textstyle {\frac {dy}{dx}}=g(x,y)}"></span> and the condition that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a2b7edf933b5c4ff37ac2bca32cb5edc0c596c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.252ex; height:2.509ex;" alt="{\displaystyle y=b}"></span> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae23950e96a955ab5b07015a168fd931d4d82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}"></span>, then there is locally a solution to this problem if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf358a54b0375e22ae5f3ab2c3e1a22c0c87e11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:2.843ex;" alt="{\displaystyle g(x,y)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\partial g}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial g}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191bf79648ec5c28dd09dc90f41f1a032dd1bdba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.708ex; height:4.176ex;" alt="{\textstyle {\frac {\partial g}{\partial x}}}"></span> are both continuous on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>. This solution exists on some interval with its center at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. The solution may not be unique. (See <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a> for other results.) </p><p>However, this only helps us with first order <a href="/wiki/Initial_value_problem" title="Initial value problem">initial value problems</a>. Suppose we had a linear initial value problem of the nth order: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}(x){\frac {d^{n}y}{dx^{n}}}+\cdots +f_{1}(x){\frac {dy}{dx}}+f_{0}(x)y=g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}(x){\frac {d^{n}y}{dx^{n}}}+\cdots +f_{1}(x){\frac {dy}{dx}}+f_{0}(x)y=g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fdbbc62fccee2cfd06d647c6dd5ab2ad034ff15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.897ex; height:5.509ex;" alt="{\displaystyle f_{n}(x){\frac {d^{n}y}{dx^{n}}}+\cdots +f_{1}(x){\frac {dy}{dx}}+f_{0}(x)y=g(x)}"></span></dd></dl> <p>such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}y(x_{0})&=y_{0},&y'(x_{0})&=y'_{0},&y''(x_{0})&=y''_{0},&\ldots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> </mtd> <mtd> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>″</mo> </msubsup> <mo>,</mo> </mtd> <mtd> <mo>…</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y(x_{0})&=y_{0},&y'(x_{0})&=y'_{0},&y''(x_{0})&=y''_{0},&\ldots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0a086a01f6d1d23e093500b3b426d23f100aa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.209ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}y(x_{0})&=y_{0},&y'(x_{0})&=y'_{0},&y''(x_{0})&=y''_{0},&\ldots \end{aligned}}}"></span></dd></dl> <p>For any nonzero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86f950ad1f8135017742e12a8da5e2367bfdc1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.497ex; height:2.843ex;" alt="{\displaystyle f_{n}(x)}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{0},f_{1},\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f_{0},f_{1},\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400e2f5ca9c6dc7392700222e59f15cff1e716c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.503ex; height:2.843ex;" alt="{\displaystyle \{f_{0},f_{1},\ldots \}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> are continuous on some interval containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> exists and is unique.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=10" title="Edit section: Related concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <a href="/wiki/Delay_differential_equation" title="Delay differential equation">delay differential equation</a> (DDE) is an equation for a function of a single variable, usually called <b>time</b>, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.</li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equations</a> may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains <a href="/wiki/Integral" title="Integral">integrals</a>.</li> <li>An <a href="/wiki/Integro-differential_equation" title="Integro-differential equation">integro-differential equation</a> (IDE) is an equation that combines aspects of a differential equation and an integral equation.</li> <li>A <a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equation</a> (SDE) is an equation in which the unknown quantity is a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> and the equation involves some known stochastic processes, for example, the <a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a> in the case of diffusion equations.</li> <li>A <a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">stochastic partial differential equation</a> (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> and <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>.</li> <li>An ultrametric <a href="/w/index.php?title=Pseudo-differential_equation&action=edit&redlink=1" class="new" title="Pseudo-differential equation (page does not exist)">pseudo-differential equation</a> is an equation which contains <a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">p-adic numbers</a> in an <a href="/wiki/Ultrametric_space" title="Ultrametric space">ultrametric space</a>. Mathematical models that involve ultrametric pseudo-differential equations use <a href="/wiki/Pseudo-differential_operators" class="mw-redirect" title="Pseudo-differential operators">pseudo-differential operators</a> instead of <a href="/wiki/Differential_operators" class="mw-redirect" title="Differential operators">differential operators</a>.</li> <li>A <a href="/wiki/Differential_algebraic_equation" class="mw-redirect" title="Differential algebraic equation">differential algebraic equation</a> (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Connection_to_difference_equations">Connection to difference equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=11" title="Edit section: Connection to difference equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Time_scale_calculus" class="mw-redirect" title="Time scale calculus">Time scale calculus</a></div> <p>The theory of differential equations is closely related to the theory of <a href="/wiki/Difference_equations" class="mw-redirect" title="Difference equations">difference equations</a>, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=12" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The study of differential equations is a wide field in <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure</a> and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and <a href="/wiki/Engineering" title="Engineering">engineering</a>. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form</a> solutions. Instead, solutions can be approximated using <a href="/wiki/Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">numerical methods</a>. </p><p>Many fundamental laws of <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> can be formulated as differential equations. In <a href="/wiki/Biology" title="Biology">biology</a> and <a href="/wiki/Economics" title="Economics">economics</a>, differential equations are used to <a href="/wiki/Mathematical_modelling" class="mw-redirect" title="Mathematical modelling">model</a> the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a>, the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a>, is governed by another second-order partial differential equation, the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a>. It turns out that many <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> processes, while seemingly different, are described by the same equation; the <a href="/wiki/Black%E2%80%93Scholes" class="mw-redirect" title="Black–Scholes">Black–Scholes</a> equation in finance is, for instance, related to the heat equation. </p><p>The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See <a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Software">Software</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=13" title="Edit section: Software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some <a href="/wiki/Computer_algebra_system" title="Computer algebra system">CAS</a> software can solve differential equations. These are the commands used in the leading programs: </p> <ul><li><a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a>:<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <code>dsolve</code></li> <li><a href="/wiki/Wolfram_Mathematica" title="Wolfram Mathematica">Mathematica</a>:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <code>DSolve[]</code></li> <li><a href="/wiki/Maxima_(software)" title="Maxima (software)">Maxima</a>:<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <code>ode2(equation, y, x)</code></li> <li><a href="/wiki/SageMath" title="SageMath">SageMath</a>:<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <code>desolve()</code></li> <li><a href="/wiki/SymPy" title="SymPy">SymPy</a>:<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <code>sympy.solvers.ode.dsolve(equation)</code></li> <li><a href="/wiki/Xcas" title="Xcas">Xcas</a>:<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <code>desolve(y'=k*y,y)</code></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact differential equation</a></li> <li><a href="/wiki/Functional_differential_equation" title="Functional differential equation">Functional differential equation</a></li> <li><a href="/wiki/Initial_condition" title="Initial condition">Initial condition</a></li> <li><a href="/wiki/Integral_equations" class="mw-redirect" title="Integral equations">Integral equations</a></li> <li><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li> <li><a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li> <li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a> on existence and uniqueness of solutions</li> <li><a href="/wiki/Recurrence_relation" title="Recurrence relation">Recurrence relation</a>, also known as 'difference equation'</li> <li><a href="/wiki/Abstract_differential_equation" title="Abstract differential equation">Abstract differential equation</a></li> <li><a href="/wiki/System_of_differential_equations" title="System of differential equations">System of differential equations</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Zill2012-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zill2012_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDennis_G._Zill2012" class="citation book cs1">Dennis G. Zill (15 March 2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pasKAAAAQBAJ&q=%22ordinary+differential%22"><i>A First Course in Differential Equations with Modeling Applications</i></a>. Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-285-40110-2" title="Special:BookSources/978-1-285-40110-2"><bdi>978-1-285-40110-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Differential+Equations+with+Modeling+Applications&rft.pub=Cengage+Learning&rft.date=2012-03-15&rft.isbn=978-1-285-40110-2&rft.au=Dennis+G.+Zill&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpasKAAAAQBAJ%26q%3D%2522ordinary%2Bdifferential%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernoulli1695" class="citation cs2"><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Bernoulli, Jacob</a> (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis", <i><a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Eruditorum&rft.atitle=Explicationes%2C+Annotationes+%26+Additiones+ad+ea%2C+quae+in+Actis+sup.+de+Curva+Elastica%2C+Isochrona+Paracentrica%2C+%26+Velaria%2C+hinc+inde+memorata%2C+%26+paratim+controversa+legundur%3B+ubi+de+Linea+mediarum+directionum%2C+alliisque+novis&rft.date=1695&rft.aulast=Bernoulli&rft.aufirst=Jacob&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHairerNørsettWanner1993" class="citation cs2">Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), <i>Solving ordinary differential equations I: Nonstiff problems</i>, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-56670-0" title="Special:BookSources/978-3-540-56670-0"><bdi>978-3-540-56670-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solving+ordinary+differential+equations+I%3A+Nonstiff+problems&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=978-3-540-56670-0&rft.aulast=Hairer&rft.aufirst=Ernst&rft.au=N%C3%B8rsett%2C+Syvert+Paul&rft.au=Wanner%2C+Gerhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrasier1983" class="citation journal cs1">Frasier, Craig (July 1983). <a rel="nofollow" class="external text" href="http://homes.chass.utoronto.ca/~cfraser/vibration.pdf">"Review of <i>The evolution of dynamics, vibration theory from 1687 to 1742</i>, by John T. Cannon and Sigalia Dostrovsky"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the American Mathematical Society</i>. New Series. <b>9</b> (1).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Review+of+The+evolution+of+dynamics%2C+vibration+theory+from+1687+to+1742%2C+by+John+T.+Cannon+and+Sigalia+Dostrovsky&rft.volume=9&rft.issue=1&rft.date=1983-07&rft.aulast=Frasier&rft.aufirst=Craig&rft_id=http%3A%2F%2Fhomes.chass.utoronto.ca%2F~cfraser%2Fvibration.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWheelerCrummett1987" class="citation journal cs1">Wheeler, Gerard F.; Crummett, William P. (1987). "The Vibrating String Controversy". <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">Am. J. Phys.</a></i> <b>55</b> (1): 33–37. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987AmJPh..55...33W">1987AmJPh..55...33W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.15311">10.1119/1.15311</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Am.+J.+Phys.&rft.atitle=The+Vibrating+String+Controversy&rft.volume=55&rft.issue=1&rft.pages=33-37&rft.date=1987&rft_id=info%3Adoi%2F10.1119%2F1.15311&rft_id=info%3Abibcode%2F1987AmJPh..55...33W&rft.aulast=Wheeler&rft.aufirst=Gerard+F.&rft.au=Crummett%2C+William+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">For a special collection of the 9 groundbreaking papers by the three authors, see <a rel="nofollow" class="external text" href="http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage=">First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200209023122/http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage=">Archived</a> 2020-02-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">For de Lagrange's contributions to the acoustic wave equation, can consult <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18">Acoustics: An Introduction to Its Physical Principles and Applications</a> Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</span> </li> <li id="cite_note-Speiser-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Speiser_9-0">^</a></b></span> <span class="reference-text">Speiser, David. <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=9uf97reZZCUC&pg=PA191">Discovering the Principles of Mechanics 1600-1800</a></i>, p. 191 (Basel: Birkhäuser, 2008).</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFourier1822" class="citation book cs1 cs1-prop-foreign-lang-source">Fourier, Joseph (1822). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_TDQJAAAAIAAJ"><i>Théorie analytique de la chaleur</i></a> (in French). Paris: Firmin Didot Père et Fils. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/2688081">2688081</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+analytique+de+la+chaleur&rft.place=Paris&rft.pub=Firmin+Didot+P%C3%A8re+et+Fils&rft.date=1822&rft_id=info%3Aoclcnum%2F2688081&rft.aulast=Fourier&rft.aufirst=Joseph&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_TDQJAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyceDiPrima1967" class="citation book cs1">Boyce, William E.; DiPrima, Richard C. (1967). <i>Elementary Differential Equations and Boundary Value Problems</i> (4th ed.). John Wiley & Sons. p. 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Differential+Equations+and+Boundary+Value+Problems&rft.pages=3&rft.edition=4th&rft.pub=John+Wiley+%26+Sons&rft.date=1967&rft.aulast=Boyce&rft.aufirst=William+E.&rft.au=DiPrima%2C+Richard+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="/wiki/Eric_W_Weisstein" class="mw-redirect" title="Eric W Weisstein">Weisstein, Eric W</a>. "Ordinary Differential Equation Order." From <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>--A Wolfram Web Resource. <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html">http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx">Order and degree of a differential equation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160401070512/http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx">Archived</a> 2016-04-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, accessed Dec 2015.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElias_Loomis1887" class="citation book cs1">Elias Loomis (1887). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DTI4AQAAMAAJ"><i>Elements of the Differential and Integral Calculus</i></a> (revised ed.). Harper & Bros. p. 247.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+the+Differential+and+Integral+Calculus&rft.pages=247&rft.edition=revised&rft.pub=Harper+%26+Bros.&rft.date=1887&rft.au=Elias+Loomis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDTI4AQAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DTI4AQAAMAAJ&pg=PA247">Extract of page 247</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZill2001" class="citation book cs1">Zill, Dennis G. (2001). <i>A First Course in Differential Equations</i> (5th ed.). Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-37388-7" title="Special:BookSources/0-534-37388-7"><bdi>0-534-37388-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Differential+Equations&rft.edition=5th&rft.pub=Brooks%2FCole&rft.date=2001&rft.isbn=0-534-37388-7&rft.aulast=Zill&rft.aufirst=Dennis+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.maplesoft.com/support/help/Maple/view.aspx?path=dsolve">"dsolve - Maple Programming Help"</a>. <i>www.maplesoft.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.maplesoft.com&rft.atitle=dsolve+-+Maple+Programming+Help&rft_id=https%3A%2F%2Fwww.maplesoft.com%2Fsupport%2Fhelp%2FMaple%2Fview.aspx%3Fpath%3Ddsolve&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/DSolve.html">"DSolve - Wolfram Language Documentation"</a>. <i>www.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-06-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.wolfram.com&rft.atitle=DSolve+-+Wolfram+Language+Documentation&rft_id=https%3A%2F%2Freference.wolfram.com%2Flanguage%2Fref%2FDSolve.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchelter" class="citation web cs1"><a href="/wiki/Bill_Schelter" title="Bill Schelter">Schelter, William F.</a> Gaertner, Boris (ed.). <a rel="nofollow" class="external text" href="https://maxima.sourceforge.io/docs/tutorial/en/gaertner-tutorial-revision/Pages/ODE0001.htm">"Differential Equations - Symbolic Solutions"</a>. <i>The Computer Algebra Program Maxima - a Tutorial (in Maxima documentation on <a href="/wiki/SourceForge" title="SourceForge">SourceForge</a>)</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221004221208/https://maxima.sourceforge.io/docs/tutorial/en/gaertner-tutorial-revision/Pages/ODE0001.htm">Archived</a> from the original on 2022-10-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Computer+Algebra+Program+Maxima+-+a+Tutorial+%28in+Maxima+documentation+on+SourceForge%29&rft.atitle=Differential+Equations+-+Symbolic+Solutions&rft.aulast=Schelter&rft.aufirst=William+F.&rft_id=https%3A%2F%2Fmaxima.sourceforge.io%2Fdocs%2Ftutorial%2Fen%2Fgaertner-tutorial-revision%2FPages%2FODE0001.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://doc.sagemath.org/html/en/tutorial/tour_algebra.html">"Basic Algebra and Calculus — Sage Tutorial v9.0"</a>. <i>doc.sagemath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=doc.sagemath.org&rft.atitle=Basic+Algebra+and+Calculus+%E2%80%94+Sage+Tutorial+v9.0&rft_id=http%3A%2F%2Fdoc.sagemath.org%2Fhtml%2Fen%2Ftutorial%2Ftour_algebra.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.sympy.org/latest/modules/solvers/ode.html">"ODE"</a>. <i>SymPy 1.11 documentation</i>. 2022-08-22. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220926154502/https://docs.sympy.org/latest/modules/solvers/ode.html">Archived</a> from the original on 2022-09-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=SymPy+1.11+documentation&rft.atitle=ODE&rft.date=2022-08-22&rft_id=https%3A%2F%2Fdocs.sympy.org%2Flatest%2Fmodules%2Fsolvers%2Fode.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-fourier.ujf-grenoble.fr/~parisse/giac/cascmd_en.pdf">"Symbolic algebra and Mathematics with Xcas"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Symbolic+algebra+and+Mathematics+with+Xcas&rft_id=http%3A%2F%2Fwww-fourier.ujf-grenoble.fr%2F~parisse%2Fgiac%2Fcascmd_en.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Differential_equation&action=edit&section=16" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbottNeill2003" class="citation book cs1">Abbott, P.; Neill, H. (2003). <i>Teach Yourself Calculus</i>. pp. 266–277.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teach+Yourself+Calculus&rft.pages=266-277&rft.date=2003&rft.aulast=Abbott&rft.aufirst=P.&rft.au=Neill%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlanchardDevaneyHall2006" class="citation book cs1">Blanchard, P.; <a href="/wiki/Robert_L._Devaney" title="Robert L. Devaney">Devaney, R. L.</a>; Hall, G. R. (2006). <i>Differential Equations</i>. Thompson.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Equations&rft.pub=Thompson&rft.date=2006&rft.aulast=Blanchard&rft.aufirst=P.&rft.au=Devaney%2C+R.+L.&rft.au=Hall%2C+G.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyceDiPrimaMeade2017" class="citation book cs1">Boyce, W.; <a href="/wiki/Richard_DiPrima" title="Richard DiPrima">DiPrima, R.</a>; Meade, D. (2017). <i>Elementary Differential Equations and Boundary Value Problems</i>. Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Differential+Equations+and+Boundary+Value+Problems&rft.pub=Wiley&rft.date=2017&rft.aulast=Boyce&rft.aufirst=W.&rft.au=DiPrima%2C+R.&rft.au=Meade%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoddingtonLevinson1955" class="citation book cs1">Coddington, E. A.; Levinson, N. (1955). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryofordinary00codd"><i>Theory of Ordinary Differential Equations</i></a></span>. McGraw-Hill.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Ordinary+Differential+Equations&rft.pub=McGraw-Hill&rft.date=1955&rft.aulast=Coddington&rft.aufirst=E.+A.&rft.au=Levinson%2C+N.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryofordinary00codd&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFInce1956" class="citation book cs1">Ince, E. L. (1956). <i>Ordinary Differential Equations</i>. Dover.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinary+Differential+Equations&rft.pub=Dover&rft.date=1956&rft.aulast=Ince&rft.aufirst=E.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson1913" class="citation book cs1">Johnson, W. (1913). <a rel="nofollow" class="external text" href="http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001"><i>A Treatise on Ordinary and Partial Differential Equations</i></a>. John Wiley and Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Ordinary+and+Partial+Differential+Equations&rft.pub=John+Wiley+and+Sons&rft.date=1913&rft.aulast=Johnson&rft.aufirst=W.&rft_id=http%3A%2F%2Fwww.hti.umich.edu%2Fcgi%2Fb%2Fbib%2Fbibperm%3Fq1%3Dabv5010.0001.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span> In <a rel="nofollow" class="external text" href="http://hti.umich.edu/u/umhistmath/">University of Michigan Historical Math Collection</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolyaninZaitsev2003" class="citation book cs1">Polyanin, A. D.; Zaitsev, V. F. (2003). <i>Handbook of Exact Solutions for Ordinary Differential Equations</i> (2nd ed.). Boca Raton: Chapman & Hall/CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-297-2" title="Special:BookSources/1-58488-297-2"><bdi>1-58488-297-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Exact+Solutions+for+Ordinary+Differential+Equations&rft.place=Boca+Raton&rft.edition=2nd&rft.pub=Chapman+%26+Hall%2FCRC+Press&rft.date=2003&rft.isbn=1-58488-297-2&rft.aulast=Polyanin&rft.aufirst=A.+D.&rft.au=Zaitsev%2C+V.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPorter1978" class="citation book cs1">Porter, R. I. (1978). "XIX Differential Equations". <i>Further Elementary Analysis</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=XIX+Differential+Equations&rft.btitle=Further+Elementary+Analysis&rft.date=1978&rft.aulast=Porter&rft.aufirst=R.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTeschl2012" class="citation book cs1"><a href="/wiki/Gerald_Teschl" title="Gerald Teschl">Teschl, Gerald</a> (2012). <a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~gerald/ftp/book-ode/"><i>Ordinary Differential Equations and Dynamical Systems</i></a>. <a href="/wiki/Providence,_Rhode_Island" title="Providence, Rhode Island">Providence</a>: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8328-0" title="Special:BookSources/978-0-8218-8328-0"><bdi>978-0-8218-8328-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinary+Differential+Equations+and+Dynamical+Systems&rft.place=Providence&rft.pub=American+Mathematical+Society&rft.date=2012&rft.isbn=978-0-8218-8328-0&rft.aulast=Teschl&rft.aufirst=Gerald&rft_id=https%3A%2F%2Fwww.mat.univie.ac.at%2F~gerald%2Fftp%2Fbook-ode%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifferential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaniel_Zwillinger2014" class="citation book cs1">Daniel Zwillinger (12 May 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=n7TiBQAAQBAJ&q=%22Handbook+of+Differential+Equations%22"><i>Handbook of Differential Equations</i></a>. 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Mathematics</li> <li><a rel="nofollow" class="external text" href="http://www.fioravante.patrone.name/mat/u-u/en/differential_equations_intro.htm">Introduction to modeling via differential equations</a> Introduction to modeling by means of differential equations, with critical remarks.</li> <li><a rel="nofollow" class="external text" href="http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode">Mathematical Assistant on Web</a> Symbolic ODE tool, using <a href="/wiki/Maxima_(software)" title="Maxima (software)">Maxima</a></li> <li><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/solutions/ode.htm">Exact Solutions of Ordinary Differential Equations</a></li> <li><a rel="nofollow" class="external text" href="http://www.hedengren.net/research/models.htm">Collection of ODE and DAE models of physical systems</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081219003453/http://www.hedengren.net/research/models.htm">Archived</a> 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href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical 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science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, 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class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations_topics" title="Template:Differential equations topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations_topics" title="Template talk:Differential equations topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations_topics" title="Special:EditPage/Template:Differential equations topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Differential_equations" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classification</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Operations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation for differentiation</a></li> <li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li> <li><a href="/wiki/Holonomic_function" title="Holonomic function">Holonomic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Attributes of variables</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a></li> <li><a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li> <li><a class="mw-selflink selflink">Coupled</a></li> <li><a class="mw-selflink selflink">Decoupled</a></li> <li><a class="mw-selflink selflink">Order</a></li> <li><a class="mw-selflink selflink">Degree</a></li> <li><a href="/wiki/Autonomous_system_(mathematics)" title="Autonomous system (mathematics)">Autonomous</a></li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact differential equation</a></li> <li><a href="/wiki/Jet_bundle#Partial_differential_equations" title="Jet bundle">On jet bundles</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Relation to processes</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference</a> (discrete analogue)</li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solutions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Existence/uniqueness</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic stability</a></li> <li><a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series solutions</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> solutions</li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_mathematical_jargon#Proof_techniques" class="mw-redirect" title="List of mathematical jargon">Inspection</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a></li> <li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Method of undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Euler_method" title="Euler method">Euler method</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></li> <li><a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson method</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></li> <li><a href="/wiki/List_of_linear_ordinary_differential_equations" title="List of linear ordinary differential equations">List of linear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_ordinary_differential_equations" title="List of nonlinear ordinary differential equations">List of nonlinear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_partial_differential_equations" title="List of nonlinear partial differential equations">List of nonlinear partial differential equations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Ernst_Leonard_Lindel%C3%B6f" title="Ernst Leonard Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Sofya_Kovalevskaya" title="Sofya Kovalevskaya">Sofya Kovalevskaya</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_topics_in_mathematical_analysis" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Analysis-footer" 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