Fattore di integrazione

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Consente inoltre di rendere <a href="/wiki/Differenziale_esatto" title="Differenziale esatto">esatto</a> un differenziale non esatto, in modo che sia possibile integrarlo ottenendo un <a href="/wiki/Campo_scalare" title="Campo scalare">campo scalare</a>. Ad esempio in <a href="/wiki/Termodinamica" title="Termodinamica">termodinamica</a> la moltiplicazione per un fattore di integrazione permette di rendere il <a href="/wiki/Calore" title="Calore">calore</a> un differenziale esatto. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="it" dir="ltr"><h2 id="mw-toc-heading">Indice</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Equazione_differenziale_lineare_del_primo_ordine"><span class="tocnumber">1</span> <span class="toctext">Equazione differenziale lineare del primo ordine</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Esempio"><span class="tocnumber">1.1</span> <span class="toctext">Esempio</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-3"><a href="#Uso_generale"><span class="tocnumber">2</span> <span class="toctext">Uso generale</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Note"><span class="tocnumber">3</span> <span class="toctext">Note</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Bibliografia"><span class="tocnumber">4</span> <span class="toctext">Bibliografia</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Voci_correlate"><span class="tocnumber">5</span> <span class="toctext">Voci correlate</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Equazione_differenziale_lineare_del_primo_ordine">Equazione differenziale lineare del primo ordine</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fattore_di_integrazione&veaction=edit&section=1" title="Modifica la sezione Equazione differenziale lineare del primo ordine" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fattore_di_integrazione&action=edit&section=1" title="Edit section's source code: Equazione differenziale lineare del primo ordine"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Equazione_differenziale_lineare" title="Equazione differenziale lineare">Equazione differenziale lineare</a></b>.</span></div> </div> <p>Si consideri un'<a href="/wiki/Equazione_differenziale_ordinaria" title="Equazione differenziale ordinaria">equazione differenziale ordinaria</a> lineare del primo ordine: </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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'+P(x)y=Q(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f547fb4f66c5d0ac5204869367a4854edb16251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.801ex; height:3.009ex;" alt="{\displaystyle y'+P(x)y=Q(x)}"></span></dd></dl> <p>Il fattore di integrazione per una tale equazione è una funzione <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> data da:<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> </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 P(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>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x)=e^{\int P(x')dx'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87729e3dc1f711995d7f1cebeb975c002d328a4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.702ex; height:3.343ex;" alt="{\displaystyle M(x)=e^{\int P(x')dx'}}"></span></dd></dl> <p>che moltiplicata per tutti i termini della relazione: </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'e^{\int P(x')dx'}+P(x)e^{\int P(x')dx'}y=Q(x)e^{\int P(x')dx'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mi>y</mi> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'e^{\int P(x')dx'}+P(x)e^{\int P(x')dx'}y=Q(x)e^{\int P(x')dx'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/712a6bad9711e9ab548e7cc1300c8cdb1de5b4b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.868ex; height:3.343ex;" alt="{\displaystyle y'e^{\int P(x')dx'}+P(x)e^{\int P(x')dx'}y=Q(x)e^{\int P(x')dx'}}"></span></dd></dl> <p>rende il membro di sinistra, attraverso la <a href="/wiki/Regola_del_prodotto" title="Regola del prodotto">regola del prodotto</a> invertita, esprimibile come una singola derivata rispetto a <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle 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 y'e^{\int P(x')dx'}+P(x)e^{\int P(x')dx'}y={\frac {d}{dx}}\left(ye^{\int P(x')dx'}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'e^{\int P(x')dx'}+P(x)e^{\int P(x')dx'}y={\frac {d}{dx}}\left(ye^{\int P(x')dx'}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/833bc49da3afba92952983e6f2e3b5f5324c3e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.591ex; height:5.509ex;" alt="{\displaystyle y'e^{\int P(x')dx'}+P(x)e^{\int P(x')dx'}y={\frac {d}{dx}}\left(ye^{\int P(x')dx'}\right)}"></span></dd></dl> <p>sicché l'equazione si semplifica nel seguente modo: </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 {d}{dx}}\left(ye^{\int P(x')dx'}\right)=Q(x)e^{\int P(x')dx'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\left(ye^{\int P(x')dx'}\right)=Q(x)e^{\int P(x')dx'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df3bc35642095a63f3da804f727fb0431fc0aed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.82ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\left(ye^{\int P(x')dx'}\right)=Q(x)e^{\int P(x')dx'}}"></span></dd></dl> <p>Integrando allora rispetto a <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> si ha: </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 ye^{\int P(x')dx'}=\int Q(x)e^{\int P(x')dx'}dx+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>=</mo> <mo>∫</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ye^{\int P(x')dx'}=\int Q(x)e^{\int P(x')dx'}dx+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab21e4c287fa0d925c9e1b9fd363f13d22f299f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.009ex; height:5.676ex;" alt="{\displaystyle ye^{\int P(x')dx'}=\int Q(x)e^{\int P(x')dx'}dx+C}"></span></dd></dl> <p>(dove <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> è una costante arbitraria) e spostando l'esponenziale al membro di destra si ottiene una soluzione generale della ODE: </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=e^{-\int P(x')dx'}\int Q(x)e^{\int P(x')dx'}dx+Ce^{-\int P(x')dx'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mo>∫</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>C</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=e^{-\int P(x')dx'}\int Q(x)e^{\int P(x')dx'}dx+Ce^{-\int P(x')dx'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ad8dc9b22a0fb0841f6c8cb896449b1c23a48c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.75ex; height:5.676ex;" alt="{\displaystyle y=e^{-\int P(x')dx'}\int Q(x)e^{\int P(x')dx'}dx+Ce^{-\int P(x')dx'}}"></span></dd></dl> <p>Se l'equazione è omogenea, ovvero <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 Q(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691acb6154b0b716ac8c916b1a57d34490538bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.238ex; height:2.843ex;" alt="{\displaystyle Q(x)=0}"></span>, si ha: </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 {C}{e^{\int P(x')dx'}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {C}{e^{\int P(x')dx'}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1baebfbc88b8ee6fdeddec0d0417a3861407bbe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.759ex; height:5.843ex;" alt="{\displaystyle y={\frac {C}{e^{\int P(x')dx'}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Esempio">Esempio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fattore_di_integrazione&veaction=edit&section=2" title="Modifica la sezione Esempio" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fattore_di_integrazione&action=edit&section=2" title="Edit section's source code: Esempio"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data l'equazione differenziale: </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>in tal caso <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 P(x)={\frac {-2}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</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 P(x)={\frac {-2}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e490844d5c14c5aa013fc512aa6132cb8057d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.79ex; height:5.176ex;" alt="{\displaystyle P(x)={\frac {-2}{x}}}"></span>, poiché: </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 P(x)\,dx}=e^{\int {\frac {-2}{x}}\,dx}=e^{-2\ln x}={(e^{\ln x})}^{-2}=x^{-2}={\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> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∫</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> </msup> <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> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <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)=e^{\int P(x)\,dx}=e^{\int {\frac {-2}{x}}\,dx}=e^{-2\ln x}={(e^{\ln x})}^{-2}=x^{-2}={\frac {1}{x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ffb26e1b8dcb0766fb6001e61d903d01dd6881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:61.493ex; height:5.509ex;" alt="{\displaystyle M(x)=e^{\int P(x)\,dx}=e^{\int {\frac {-2}{x}}\,dx}=e^{-2\ln x}={(e^{\ln x})}^{-2}=x^{-2}={\frac {1}{x^{2}}}}"></span></dd></dl> <p>Moltiplicando 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> si ottiene: </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}}}={\frac {y'x^{3}-2x^{2}y}{x^{5}}}={\frac {x(y'x^{2}-2xy)}{x^{5}}}={\frac {y'x^{2}-2xy}{x^{4}}}=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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</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}}}={\frac {y'x^{3}-2x^{2}y}{x^{5}}}={\frac {x(y'x^{2}-2xy)}{x^{5}}}={\frac {y'x^{2}-2xy}{x^{4}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26225c093729d2e707df10f0e0d83f5d7943d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:61.69ex; height:6.176ex;" alt="{\displaystyle {\frac {y'}{x^{2}}}-{\frac {2y}{x^{3}}}={\frac {y'x^{3}-2x^{2}y}{x^{5}}}={\frac {x(y'x^{2}-2xy)}{x^{5}}}={\frac {y'x^{2}-2xy}{x^{4}}}=0}"></span></dd></dl> <p>e dalla <a href="/wiki/Regola_del_quoziente" title="Regola del quoziente">regola del quoziente</a> invertita: </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 \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>ovvero: </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>che fornisce: </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\left(x\right)=Cx^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <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\left(x\right)=Cx^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68080afb723ebfb8ed37a1a544ca70b36da996f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.93ex; height:3.176ex;" alt="{\displaystyle y\left(x\right)=Cx^{2}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Uso_generale">Uso generale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fattore_di_integrazione&veaction=edit&section=3" title="Modifica la sezione Uso generale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fattore_di_integrazione&action=edit&section=3" title="Edit section's source code: Uso generale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si consideri l'equazione non lineare del secondo ordine: </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 {d^{2}y}{dt^{2}}}=Ay^{2/3}}"> <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>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>A</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dt^{2}}}=Ay^{2/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82a4a72314fc7307a6d6d2e2c34e581a096725f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.964ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}y}{dt^{2}}}=Ay^{2/3}}"></span></dd></dl> <p>e sia <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 {\tfrac {dy}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {dy}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/481a6b80e8ead2ac7c4773f4b0484b2efe6efe28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.513ex; height:4.176ex;" alt="{\displaystyle {\tfrac {dy}{dt}}}"></span> un fattore di integrazione: </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 {d^{2}y}{dt^{2}}}{\frac {dy}{dt}}=Ay^{2/3}{\frac {dy}{dt}}}"> <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>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>A</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dt^{2}}}{\frac {dy}{dt}}=Ay^{2/3}{\frac {dy}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd2831f1694fd65cbeb64e39d13ad55af72c702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.379ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}y}{dt^{2}}}{\frac {dy}{dt}}=Ay^{2/3}{\frac {dy}{dt}}}"></span></dd></dl> <p>Attraverso la <a href="/wiki/Regola_della_catena" title="Regola della catena">regola della catena</a> si possono esprimere entrambi i membri come una derivata: </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 {d}{dt}}\left({\frac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}\right)={\frac {d}{dt}}\left(A{\frac {3}{5}}y^{5/3}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}\right)={\frac {d}{dt}}\left(A{\frac {3}{5}}y^{5/3}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026f6fbe6bf6f42937ec9ba8256aa028126a3e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.04ex; height:7.509ex;" alt="{\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}\right)={\frac {d}{dt}}\left(A{\frac {3}{5}}y^{5/3}\right)}"></span></dd></dl> <p>Quindi: </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 \left({\frac {dy}{dt}}\right)^{2}={\frac {6A}{5}}y^{5/3}+C_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mi>A</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {dy}{dt}}\right)^{2}={\frac {6A}{5}}y^{5/3}+C_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d7133e2e903390ac86e98a0c110aa52197abc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.938ex; height:6.509ex;" alt="{\displaystyle \left({\frac {dy}{dt}}\right)^{2}={\frac {6A}{5}}y^{5/3}+C_{0}}"></span></dd></dl> <p>Da cui si ottiene, separando le variabili: </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 \int {\frac {dy}{\sqrt {{\frac {6A}{5}}y^{5/3}+C_{0}}}}=t+C_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mi>A</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mi>t</mi> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {dy}{\sqrt {{\frac {6A}{5}}y^{5/3}+C_{0}}}}=t+C_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a04d17824c6a37c72d06bff9568d7575712c55a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:27.541ex; height:8.176ex;" alt="{\displaystyle \int {\frac {dy}{\sqrt {{\frac {6A}{5}}y^{5/3}+C_{0}}}}=t+C_{1}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fattore_di_integrazione&veaction=edit&section=4" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fattore_di_integrazione&action=edit&section=4" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Vladimir Ivanovič Smirnov, <span style="font-style:italic;">Corso di matematica superiore</span>, vol. 2, Editori Riuniti University Press, 2011 <abbr title="Data di edizione originale">[1977]</abbr>, p. 258, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-6473-217-6" title="Speciale:RicercaISBN/978-88-6473-217-6">978-88-6473-217-6</a>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/IntegratingFactor.html"><span style="font-style:italic;">MathWorld - Integrating Factor</span></a>, su <span style="font-style:italic;">mathworld.wolfram.com</span>. <small>URL consultato il 31-01-2013</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fattore_di_integrazione&veaction=edit&section=5" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fattore_di_integrazione&action=edit&section=5" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Adams, R. A. Calculus: <i>A Complete Course, 4th ed</i>. Reading, MA: Addison Wesley, 1999.</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Morse, P. M. and Feshbach, H. <i>Methods of Theoretical Physics, Part I</i>. New York: McGraw-Hill, pp. 526-529, 1953.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fattore_di_integrazione&veaction=edit&section=6" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Fattore_di_integrazione&action=edit&section=6" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Differenziale_esatto" title="Differenziale esatto">Differenziale esatto</a></li> <li><a href="/wiki/Equazione_differenziale_lineare" title="Equazione differenziale lineare">Equazione differenziale lineare</a></li> <li><a href="/wiki/Equazione_differenziale_ordinaria" title="Equazione differenziale ordinaria">Equazione 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> <h3 id="p-Comunità-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Comunità</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Portale:Comunit%C3%A0" title="Descrizione del progetto, cosa puoi fare, dove trovare le cose"><span>Portale Comunità</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:Bar"><span>Bar</span></a></li><li id="n-wikipediano" class="mw-list-item"><a href="/wiki/Wikipedia:Wikipediano"><span>Il Wikipediano</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_it.wikipedia.org&uselang=it" title="Sostienici"><span>Fai una donazione</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Wikipedia:Contatti"><span>Contatti</span></a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Strumenti</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Fattore_di_integrazione" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Fattore_di_integrazione" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q"><span>Pagine speciali</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Fattore_di_integrazione&oldid=127720706" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Fattore_di_integrazione&action=info" title="Ulteriori informazioni su questa pagina"><span>Informazioni pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Fattore_di_integrazione&id=127720706&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina"><span>Cita questa voce</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciale:UrlShortener&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FFattore_di_integrazione"><span>Ottieni URL breve</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speciale:QrCode&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FFattore_di_integrazione"><span>Scarica codice QR</span></a></li> </ul> </div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Stampa/esporta</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciale:Libro&bookcmd=book_creator&referer=Fattore+di+integrazione"><span>Crea un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciale:DownloadAsPdf&page=Fattore_di_integrazione&action=show-download-screen"><span>Scarica come PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Fattore_di_integrazione&printable=yes" title="Versione stampabile di questa pagina [p]" accesskey="p"><span>Versione stampabile</span></a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In altri progetti</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2734005" title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g"><span>Elemento Wikidata</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In altre lingue</span> </h3> <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 - arabo" lang="ar" hreflang="ar" data-title="عامل تكاملي" data-language-autonym="العربية" data-language-local-name="arabo" 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 - catalano" lang="ca" hreflang="ca" data-title="Factor d'integració" data-language-autonym="Català" data-language-local-name="catalano" 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 - ceco" lang="cs" hreflang="cs" data-title="Integrační faktor" data-language-autonym="Čeština" data-language-local-name="ceco" 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 - inglese" lang="en" hreflang="en" data-title="Integrating factor" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Integralanta_faktoro" title="Integralanta faktoro - esperanto" lang="eo" hreflang="eo" data-title="Integralanta faktoro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factor_integrador" title="Factor integrador - spagnolo" lang="es" hreflang="es" data-title="Factor integrador" data-language-autonym="Español" data-language-local-name="spagnolo" 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 - francese" lang="fr" hreflang="fr" data-title="Facteur intégrant" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</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 - giapponese" lang="ja" hreflang="ja" data-title="積分因子" data-language-autonym="日本語" data-language-local-name="giapponese" 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 - khmer" lang="km" hreflang="km" data-title="កត្តាអាំងតេក្រាល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" 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 - coreano" lang="ko" hreflang="ko" data-title="적분인자" data-language-autonym="한국어" data-language-local-name="coreano" 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 - lituano" lang="lt" hreflang="lt" data-title="Integruojantis daugiklis" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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 - olandese" lang="nl" hreflang="nl" data-title="Integrerende factor" data-language-autonym="Nederlands" data-language-local-name="olandese" 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 - polacco" lang="pl" hreflang="pl" data-title="Czynnik całkujący" data-language-autonym="Polski" data-language-local-name="polacco" 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 - portoghese" lang="pt" hreflang="pt" data-title="Fator integrante" data-language-autonym="Português" data-language-local-name="portoghese" 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 - rumeno" lang="ro" hreflang="ro" data-title="Factor integrant" data-language-autonym="Română" data-language-local-name="rumeno" 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 - albanese" lang="sq" hreflang="sq" data-title="Faktori integrues" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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Fattore di integrazione - Wikipedia