Definition:Exact Differential Equation

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

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:
 * $$\frac {\partial f} {\partial x} = M, \frac {\partial f} {\partial y} = N$$

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

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