View source for Differential equation with total differential

An ordinary 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/d/d032/d032050/d0320501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

whose left-hand side is a total derivative:

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In other words, equation (1) is a differential equation with total differential if there exists a differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320503.png" /> such that

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identically with respect to all arguments. The solution of a differential equation with total differential of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320505.png" /> is reduced to solving an equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320506.png" />:

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320508.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320509.png" /> times continuously-differentiable function and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205010.png" /> be a function having continuous partial derivatives up to and including the second order. Let

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For equation (1) to be a differential equation with total differential it is sufficient that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205014.png" />, are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205015.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205016.png" /> [[#References|[1]]]. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205017.png" /> may enter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205018.png" /> in a linear manner only.

The first-order 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/d/d032/d032050/d03205019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205023.png" /> are defined and continuous in an open simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205024.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205025.png" />-plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205027.png" />, is a differential equation with total differential if and only if

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The general solution of equation (2) with total differential has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205029.png" />, where

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and the integral is taken over any rectifiable curve lying inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205031.png" /> and joining an arbitrary fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205032.png" /> with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205033.png" /> [[#References|[2]]]. Equation (2) (in the general case, an equation (1) which is linear with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205034.png" />) can, under certain conditions, be reduced to a differential equation with total differential by multiplying by an [[Integrating factor|integrating factor]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.P. Erugin,   "A general course in differential equations" , Minsk  (1972)  (In Russian)</TD></TR></table>