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Difference between revisions of "Integrating factor"

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''of an ordinary first-order differential equation
 
''of an ordinary first-order differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517101.png" /></td> </tr></table>
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$$P(x,y)dx+Q(x,y)dy=0$$
  
 
''
 
''
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517102.png" /> with the property that
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A function $\mu=\mu(x,y)\not\equiv0$ with the property that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517103.png" /></td> </tr></table>
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$$\mu(x,y)P(x,y)dx+\mu(x,y)Q(x,y)dy=0$$
  
is a [[Differential equation with total differential|differential equation with total differential]]. E.g., for the linear equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517104.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517105.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517106.png" /> is an integrating factor. If in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517107.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517108.png" /> equation
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is a [[Differential equation with total differential|differential equation with total differential]]. E.g., for the linear equation $y'+a(x)y=f(x)$, or $(a(x)y-f(x))dx+dy=0$, the function $\mu=\exp\int a(x)dx$ is an integrating factor. If in a domain $D$ where $P^2+Q^2\neq0$ equation
  
has a smooth [[General integral|general integral]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i0517109.png" />, then it has an infinite number of integrating factors. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i05171010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i05171011.png" /> have continuous partial derivatives in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i05171012.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i05171013.png" />, then any particular (non-trivial) solution of the partial differential equation
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has a smooth [[General integral|general integral]] $U(x,y)=C$, then it has an infinite number of integrating factors. If $P(x,y)$ and $Q(x,y)$ have continuous partial derivatives in a domain $D$ where $P^2+Q^2\neq0$, then any particular (non-trivial) solution of the partial differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051710/i05171014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$Q\frac{\partial u}{\partial x}-P\frac{\partial u}{\partial y}+\mu\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)=0\tag{2}$$
  
can be taken as integrating factor, see [[#References|[1]]]. However, a general method for finding solutions of (2) does not exist, and hence it is only in exceptional cases that one succeeds in finding an integrating factor for a concrete equation , cf. [[#References|[2]]].
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can be taken as integrating factor, see [[#References|[1]]]. However, a general method for finding solutions of \ref{2} does not exist, and hence it is only in exceptional cases that one succeeds in finding an integrating factor for a concrete equation , cf. [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,  "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,  "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR></table>

Revision as of 11:34, 1 August 2014

of an ordinary first-order differential equation

$$P(x,y)dx+Q(x,y)dy=0$$

A function $\mu=\mu(x,y)\not\equiv0$ with the property that

$$\mu(x,y)P(x,y)dx+\mu(x,y)Q(x,y)dy=0$$

is a differential equation with total differential. E.g., for the linear equation $y'+a(x)y=f(x)$, or $(a(x)y-f(x))dx+dy=0$, the function $\mu=\exp\int a(x)dx$ is an integrating factor. If in a domain $D$ where $P^2+Q^2\neq0$ equation

has a smooth general integral $U(x,y)=C$, then it has an infinite number of integrating factors. If $P(x,y)$ and $Q(x,y)$ have continuous partial derivatives in a domain $D$ where $P^2+Q^2\neq0$, then any particular (non-trivial) solution of the partial differential equation

$$Q\frac{\partial u}{\partial x}-P\frac{\partial u}{\partial y}+\mu\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)=0\tag{2}$$

can be taken as integrating factor, see [1]. However, a general method for finding solutions of \ref{2} does not exist, and hence it is only in exceptional cases that one succeeds in finding an integrating factor for a concrete equation , cf. [2].

References

[1] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
How to Cite This Entry:
Integrating factor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrating_factor&oldid=13624
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article