Definition:Integrating Factor

Definition

Consider the first order ordinary differential equation:

$(1): \qquad M \left({x, y}\right) + N \left({x, y}\right) \dfrac {dy} {dx} = 0$

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.

See also

• 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.

Sources

• George F. Simmons: Differential Equations (1972): $\S 9$