Definition:Integrating Factor

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

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

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

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

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.

Also known as
An integrating factor is sometimes known as an Euler multiplier, after, who first introduced such a notion.

Also see

 * Existence of Integrating Factor, in which it is shown that if an equation in the form of $(1)$ has a general solution, then it always has an integrating factor.