Integrating factor

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

Integrating factors are useful for solving ordinary differential equations that can be expressed in the form

The basic idea is to find some function ${\displaystyle M(x)}$, called the "integrating factor," which we can multiply through our differential equation in order to bring the left-hand side under a common derivative. For the canonical first-order, linear differential equation shown above, our integrating factor is chosen to be

In order to derive this, let ${\displaystyle M(x)}$ be the integrating factor of a first order, linear differential equation such that multiplication by ${\displaystyle M(x)}$ transforms a partial derivative into a total derivative, then:

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