Definition:Integrating Factor

Definition
Consider the first order ordinary differential equation:
 * $$(1) \qquad M \left({x, y}\right) + N \left({x, y}\right) \frac {dy} {dx} = 0$$

such that $$M$$ and $$N$$ are real functions of two variables which are not homogeneous functions of the same degree.

Suppose also that:
 * $$\frac {\partial M} {\partial y} \ne \frac {\partial N} {\partial x}$$

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 $$\mu \left({x, y}\right)$$ such that:
 * $$\mu \left({x, y}\right) \left({M \left({x, y}\right) + N \left({x, y}\right) \frac {dy} {dx}}\right) = 0$$

is exact.

Then the solution of $$(1)$$ can be found by the technique defined in Solution to Exact Differential Equation.

The function $$\mu \left({x, y}\right)$$ is called an integrating factor.