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


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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