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Havebla en 18 lingvo" > <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-18" 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">18 lingvoj</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%A7%D9%85%D9%84_%D8%AA%D9%83%D8%A7%D9%85%D9%84%D9%8A" title="عامل تكاملي — araba" lang="ar" hreflang="ar" data-title="عامل تكاملي" data-language-autonym="العربية" data-language-local-name="araba" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Factor_d%27integraci%C3%B3" title="Factor d'integració — kataluna" lang="ca" hreflang="ca" data-title="Factor d'integració" data-language-autonym="Català" data-language-local-name="kataluna" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Integra%C4%8Dn%C3%AD_faktor" title="Integrační faktor — ĉeĥa" lang="cs" hreflang="cs" data-title="Integrační faktor" data-language-autonym="Čeština" data-language-local-name="ĉeĥa" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Integrating_factor" title="Integrating factor — angla" lang="en" hreflang="en" data-title="Integrating factor" data-language-autonym="English" data-language-local-name="angla" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factor_integrador" title="Factor integrador — hispana" lang="es" hreflang="es" data-title="Factor integrador" data-language-autonym="Español" data-language-local-name="hispana" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Facteur_int%C3%A9grant" title="Facteur intégrant — franca" lang="fr" hreflang="fr" data-title="Facteur intégrant" data-language-autonym="Français" data-language-local-name="franca" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Fattore_di_integrazione" title="Fattore di integrazione — itala" lang="it" hreflang="it" data-title="Fattore di integrazione" data-language-autonym="Italiano" data-language-local-name="itala" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A9%8D%E5%88%86%E5%9B%A0%E5%AD%90" title="積分因子 — japana" lang="ja" hreflang="ja" data-title="積分因子" data-language-autonym="日本語" data-language-local-name="japana" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9E%8F%E1%9F%92%E1%9E%8F%E1%9E%B6%E1%9E%A2%E1%9E%B6%E1%9F%86%E1%9E%84%E1%9E%8F%E1%9F%81%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%B6%E1%9E%9B" title="កត្តាអាំងតេក្រាល — kmera" lang="km" hreflang="km" data-title="កត្តាអាំងតេក្រាល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="kmera" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%81%EB%B6%84%EC%9D%B8%EC%9E%90" title="적분인자 — korea" lang="ko" hreflang="ko" data-title="적분인자" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Integruojantis_daugiklis" title="Integruojantis daugiklis — litova" lang="lt" hreflang="lt" data-title="Integruojantis daugiklis" data-language-autonym="Lietuvių" data-language-local-name="litova" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Integrerende_factor" title="Integrerende factor — nederlanda" lang="nl" hreflang="nl" data-title="Integrerende factor" data-language-autonym="Nederlands" data-language-local-name="nederlanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Czynnik_ca%C5%82kuj%C4%85cy" title="Czynnik całkujący — pola" lang="pl" hreflang="pl" data-title="Czynnik całkujący" data-language-autonym="Polski" data-language-local-name="pola" 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/Fator_integrante" title="Fator integrante — portugala" lang="pt" hreflang="pt" data-title="Fator integrante" data-language-autonym="Português" data-language-local-name="portugala" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Factor_integrant" title="Factor integrant — rumana" lang="ro" hreflang="ro" data-title="Factor integrant" data-language-autonym="Română" data-language-local-name="rumana" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Faktori_integrues" title="Faktori integrues — albana" lang="sq" hreflang="sq" data-title="Faktori integrues" data-language-autonym="Shqip" data-language-local-name="albana" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D1%83%D0%B2%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BC%D0%BD%D0%BE%D0%B6%D0%BD%D0%B8%D0%BA" title="Інтегрувальний множник — ukraina" lang="uk" hreflang="uk" data-title="Інтегрувальний множник" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%A7%AF%E5%88%86%E5%9B%A0%E5%AD%90" title="积分因子 — ĉina" lang="zh" hreflang="zh" data-title="积分因子" data-language-autonym="中文" data-language-local-name="ĉina" class="interlanguage-link-target"><span>中文</span></a></li> </ul> 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Bonvolu <a class="external text" href="https://eo.wikipedia.org/w/index.php?title=Integralanta_faktoro&action=edit">ŝanĝi</a> la enhavon por plibonigi la artikolon.</span></td> </tr> </tbody></table> <p>En <a href="/wiki/Matematiko" title="Matematiko">matematiko</a>, oni solvas certajn <a href="/wiki/Ordinara_diferenciala_ekvacio" title="Ordinara diferenciala ekvacio">ordinarajn</a> <a href="/wiki/Diferenciala_ekvacio" title="Diferenciala ekvacio">diferencialajn ekvaciojn</a> per uzo de <b>integralanta faktoro</b>. La integralanta faktoro estas funkcio elektita ĝuste tiel ke per ĝi eblas solvi la donitan ekvacion. </p><p>Konsideru ordinaran diferencialan ekvacion de formo </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'+a(x)y=b(x)\quad \quad \quad (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'+a(x)y=b(x)\quad \quad \quad (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a358971162a16a9a05abcd5b5e7f6cf1b8644a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.384ex; height:3.009ex;" alt="{\displaystyle y'+a(x)y=b(x)\quad \quad \quad (1)}"></span></dd></dl> <p>kie <i>y = y(x)</i> estas nekonata funkcio de <i>x</i>, kaj <i>a(x)</i> kaj <i>b(x)</i> estas donitaj funkcioj. </p><p>La maniero de integralanta faktoro laboras per transformigo de la maldekstra flanko enen la formon de <a href="/wiki/Deriva%C4%B5o_de_produto" title="Derivaĵo de produto">derivaĵo de produto</a>. </p><p>Konsideri funkcion <i>M(x)</i>. Oni multipliki ambaŭ flankojn de (1) je <i>M(x)</i>: </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 M(x)y'+M(x)a(x)y=M(x)b(x)\quad \quad \quad (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)y'+M(x)a(x)y=M(x)b(x)\quad \quad \quad (2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15276b01aa6f86d106164dfaf3237b0ff12f5480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.128ex; height:3.009ex;" alt="{\displaystyle M(x)y'+M(x)a(x)y=M(x)b(x)\quad \quad \quad (2)}"></span></dd></dl> <p>Necesas ke la maldekstra flanko estu en formo de derivaĵo de produto. Fakte, se alpreni ĉi tion la maldekstra flanko povas esti reordigita kiel </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 (M(x)y)'=M(x)b(x)\quad \quad \quad (3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M(x)y)'=M(x)b(x)\quad \quad \quad (3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c27d466a819f35f1b43c0c2f4ced2fb6710894a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.986ex; height:3.009ex;" alt="{\displaystyle (M(x)y)'=M(x)b(x)\quad \quad \quad (3)}"></span></dd></dl> <p>La maldekstra flanko povas esti integralita multe pli facile per la <a href="/wiki/Fundamenta_teoremo_de_kalkulo" title="Fundamenta teoremo de kalkulo">fundamenta teoremo de kalkulo</a>, </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(x)M(x)=\int b(x)M(x)\,dx+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)M(x)=\int b(x)M(x)\,dx+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318f484353899044cdb44db2185349c4f5794ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.812ex; height:5.676ex;" alt="{\displaystyle y(x)M(x)=\int b(x)M(x)\,dx+C}"></span></dd></dl> <p>kie <i>C</i> estas <a href="/w/index.php?title=Konstanto_de_integralado&action=edit&redlink=1" class="new" title="Konstanto de integralado (paĝo ne ekzistas)">konstanto de integralado</a>. Oni povas nun solvi por <i>y(x)</i> </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(x)={\frac {\int b(x)M(x)\,dx+C}{M(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mrow> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)={\frac {\int b(x)M(x)\,dx+C}{M(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc815afc45a3ce8a63ee1ec9cb21ddcc1cd1e0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.291ex; height:6.676ex;" alt="{\displaystyle y(x)={\frac {\int b(x)M(x)\,dx+C}{M(x)}}}"></span></dd></dl> <p>Tamen, por eksplicita solvo por <i>y(x)</i> oni bezonas trovi esprimon por <i>M(x)</i>. Povas esti konkludite de (2) ke <i>M(x)</i> obeas diferencialan ekvacion </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 M'(x)-a(x)M(x)=0\quad \quad \quad (4)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M'(x)-a(x)M(x)=0\quad \quad \quad (4)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad13f6259ac9fac75f9ad62a6057ea95753ef069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.7ex; height:3.009ex;" alt="{\displaystyle M'(x)-a(x)M(x)=0\quad \quad \quad (4)\,}"></span></dd></dl> <p>Al preni <i>M(x)', dividu ambaŭ flankojn per </i>M(x)<i>:</i> </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 {\frac {M'(x)}{M(x)}}-a(x)=0\quad \quad \quad (5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {M'(x)}{M(x)}}-a(x)=0\quad \quad \quad (5)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b862f1d5eba731bad0df1a5b489ddf283644c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.568ex; height:6.509ex;" alt="{\displaystyle {\frac {M'(x)}{M(x)}}-a(x)=0\quad \quad \quad (5)}"></span></dd></dl> <p>Ekvacio (5) estas nun en formo de <a href="/w/index.php?title=Logaritma_deriva%C4%B5o&action=edit&redlink=1" class="new" title="Logaritma derivaĵo (paĝo ne ekzistas)">logaritma derivaĵo</a>. Solvo de (5) donas ke </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 M(x)=e^{\int a(x)\,dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)=e^{\int a(x)\,dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0889126bbc11a687e4db6d556ce171f8abb843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.661ex; height:3.343ex;" alt="{\displaystyle M(x)=e^{\int a(x)\,dx}}"></span></dd></dl> <p>Oni vidas ke multiplikante per <i>M(x)</i> kaj la propraĵo <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 M'(x)=a(x)M(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M'(x)=a(x)M(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae605dea6c7d8c033cb6601339f406079d571dd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.371ex; height:3.009ex;" alt="{\displaystyle M'(x)=a(x)M(x)}"></span> estis esenca en solvado de ĉi tiu diferenciala ekvacio. <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 M(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2fab18f2df9b523ef8b7b63a291317294fc708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.581ex; height:2.843ex;" alt="{\displaystyle M(x)}"></span> estas la <b>integralanta faktoro</b>. La nomo venas de la fakto ke ĝi estas integralo, kaj ĝi montras kiel <i>faktoro</i> en la ekvacio. </p> <div class="mw-heading mw-heading2"><h2 id="Ekzemplo">Ekzemplo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integralanta_faktoro&veaction=edit&section=1" title="Redakti sekcion: Ekzemplo" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Integralanta_faktoro&action=edit&section=1" title="Redakti la fontkodo de la sekcio: Ekzemplo"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Solvu la diferencialan ekvacion </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'-{\frac {2y}{x}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>y</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'-{\frac {2y}{x}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee561be858232d5488f46aa0a9239cab563f953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.101ex; height:5.343ex;" alt="{\displaystyle y'-{\frac {2y}{x}}=0}"></span></dd></dl> <p>Oni povas vidi ke en ĉi tiu okazo <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(x)={\frac {-2}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(x)={\frac {-2}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4c0251ac7e95e8a1506a81add2dee132851972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.274ex; height:5.176ex;" alt="{\displaystyle a(x)={\frac {-2}{x}}}"></span> </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 M(x)=e^{\int a(x)\,dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)=e^{\int a(x)\,dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0889126bbc11a687e4db6d556ce171f8abb843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.661ex; height:3.343ex;" alt="{\displaystyle M(x)=e^{\int a(x)\,dx}}"></span></dd></dl> <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 M(x)=e^{\int {\frac {-2}{x}}\,dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)=e^{\int {\frac {-2}{x}}\,dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24b19d79ec05ccae980e08b5f30f2bccce80c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.114ex; height:3.843ex;" alt="{\displaystyle M(x)=e^{\int {\frac {-2}{x}}\,dx}}"></span></dd></dl> <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 M(x)={\frac {1}{x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)={\frac {1}{x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/390d70fb14a594a0a95a9afdbb9704cb271dac95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.9ex; height:5.509ex;" alt="{\displaystyle M(x)={\frac {1}{x^{2}}}}"></span></dd></dl> <p>Multiplikante ambaŭ flankojn per <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 M(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2fab18f2df9b523ef8b7b63a291317294fc708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.581ex; height:2.843ex;" alt="{\displaystyle M(x)}"></span> oni ricevas ke </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 {\frac {y'}{x^{2}}}-{\frac {2y}{x^{3}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>y</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {y'}{x^{2}}}-{\frac {2y}{x^{3}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60982aace0946ad082e34f6c570908d6ec43222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.541ex; height:5.843ex;" alt="{\displaystyle {\frac {y'}{x^{2}}}-{\frac {2y}{x^{3}}}=0}"></span></dd> <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 \left({\frac {y}{x^{2}}}\right)'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {y}{x^{2}}}\right)'=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a0708192c9cd96d3489626cbc64252c3483316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.587ex; height:6.343ex;" alt="{\displaystyle \left({\frac {y}{x^{2}}}\right)'=0}"></span></dd></dl> <p>aŭ </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 {\frac {y}{x^{2}}}=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {y}{x^{2}}}=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2f9ccbd2b76dfc4ef7e7c8b8b35e530acc78f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.085ex; height:5.176ex;" alt="{\displaystyle {\frac {y}{x^{2}}}=C}"></span></dd></dl> <p>kiu donas ke </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(x)=Cx^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)=Cx^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4640a40c121da33f78495dc13d87c310b7d7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.543ex; height:3.176ex;" alt="{\displaystyle y(x)=Cx^{2}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vidu_ankaŭ"><span id="Vidu_anka.C5.AD"></span>Vidu ankaŭ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integralanta_faktoro&veaction=edit&section=2" title="Redakti sekcion: Vidu ankaŭ" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Integralanta_faktoro&action=edit&section=2" title="Redakti la fontkodo de la sekcio: Vidu ankaŭ"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Maniero_de_variacio_de_parametroj&action=edit&redlink=1" class="new" title="Maniero de variacio de parametroj (paĝo ne ekzistas)">Maniero de variacio de parametroj</a></li> <li><a href="/w/index.php?title=Ekzemploj_de_diferencialaj_ekvacioj&action=edit&redlink=1" class="new" title="Ekzemploj de diferencialaj ekvacioj (paĝo ne ekzistas)">Ekzemploj de diferencialaj ekvacioj</a></li> <li><a href="/w/index.php?title=Logaritma_deriva%C4%B5o&action=edit&redlink=1" class="new" title="Logaritma derivaĵo (paĝo ne ekzistas)">Logaritma derivaĵo</a></li> <li><a href="/wiki/Produta_regulo" class="mw-redirect" title="Produta regulo">Produta regulo</a></li> <li><a href="/w/index.php?title=Akurata_diferencialo&action=edit&redlink=1" class="new" title="Akurata diferencialo (paĝo ne ekzistas)">Akurata diferencialo</a></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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