\begin{definition}[Definition:Exact Differential Equation]
Let a first order ordinary differential equation be expressible in this form:
:$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
such that $M$ and $N$ are ''not'' homogeneous functions of the same degree.
However, suppose there happens to exist a function $\map f {x, y}$ 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 $M \rd x + N \rd y$ is called an '''exact differential''', and the differential equation is called an '''exact differential equation'''.
\end{definition}