\begin{definition}[Definition:Integrating Factor]
Consider the first order ordinary differential equation:
:$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree.
Suppose also that:
:$\dfrac {\partial M} {\partial y} \ne \dfrac {\partial N} {\partial x}$
Then from Solution to Exact Differential Equation, $(1)$ is not exact, and that method can not be used to solve it.
However, suppose we can find a real function of two variables $\map \mu {x, y}$ such that:
:$\map \mu {x, y} \paren {\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} } = 0$
is exact.
Then the solution of $(1)$ ''can'' be found by the technique defined in Solution to Exact Differential Equation.
The function $\map \mu {x, y}$ is called an '''integrating factor'''.
\end{definition}