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 $

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