Definition:Exact Differential Equation

Definition
Let a first order ordinary differential equation be expressible in this form:
 * $M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm dy} {\mathrm dx} = 0$

such that $M$ and $N$ are not homogeneous functions of the same degree.

However, suppose there happens to exist 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.

Then the expression $M \, \mathrm dx + N \, \mathrm dy$ is called an exact differential, and the differential equation is called an exact differential equation.

Also see

 * Solution to Exact Differential Equation
 * Chain Rule for Real-Valued Functions (Corollary)