# Category:Definitions/Integrating Factors

This category contains definitions related to Integrating Factors.

Related results can be found in Category:Integrating Factors.

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**.

## Pages in category "Definitions/Integrating Factors"

The following 2 pages are in this category, out of 2 total.