# Definition:Integrating Factor

## Definition

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.

## Sources

- 1958: G.E.H. Reuter:
*Elementary Differential Equations & Operators*... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.2$ The integrating factor - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.9$: Integrating Factors - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Euler multiplier** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Euler multiplier** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**Euler multiplier**