Definition:Integrating Factor

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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.

Also known as

An integrating factor is sometimes known as an Euler multiplier, after Leonhard Paul Euler, 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.
  • Results about integrating factors can be found here.

Historical Note

The technique of using an integrating factor to solve a differential equation appears to have been invented by Leonhard Paul Euler, in $1734$.

It was also independently invented by both Alexis Fontaine des Bertins and Alexis Claude Clairaut.

Some sources attribute it to Gottfried Wilhelm von Leibniz.