Definition:Exact Differential Equation

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

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 $M \, \mathrm d x + N \, \mathrm d y$ is called an exact differential, and the differential equation is called an exact differential equation.

Also presented as
An exact differential equation can also be presented as:


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

or:


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

or in differential form as:


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

or:


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

all with the same conditions on $f \left({x, y}\right)$, $\dfrac {\partial f} {\partial x}$ and $\dfrac {\partial f} {\partial y}$.

Also see

 * Solution to Exact Differential Equation