## FANDOM

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An exact differential equation is an ordinary differential equation in the form

$m(x,y)dx+n(x,y)dy=0\quad\text{if}\quad\frac{\part}{\part y}m(x,y)=\frac{\part}{\part x}n(x,y)$

If this is the case, we can assume $m(x,y)$ and $n(x,y)$ are the partial derivatives of an unknown function $\psi(x,y)$ , due to the symmetry of second order partial derivatives. The function can now be written as

$\frac{d}{dx}\psi(x,y)=0$

Since only constants have derivatives of 0, $\psi=C$ . Now $m(x,y)$ can be integrated in terms of $x$ to find $f(x)$ . Since this is a partial derivative in terms of $x$ , any functions of $y$ will be lost, so instead of adding a constant of integration, we will add $f(y)$ . $f(y)$ can be found by integrating $n(x,y)$ in terms of $y$ , yielding the final solution is:

$\int m(x,y)dx+\int n(x,y)dy=C$