Exact Differential Equation/Examples

Definition
This page gathers examples of exact differential equations of the first order:


 * $M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$

where there exists a function $f \left({x, y}\right)$ such that:
 * $\dfrac {\partial f} {\partial x} = M, \dfrac {\partial f} {\partial y} = N$

such that the second partial derivatives of $f$ exist and are continuous.

$e^y + \cos x \cos y = \left({\sin x \sin y - x e^y}\right) \dfrac {\mathrm d y} {\mathrm d x}$
=== $-\dfrac 1 y \sin \dfrac x y + \left({\dfrac x {y^2} \sin \dfrac x y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$ ===