Integrating factor

Definition

Let $I$ be an open interval, and let $p, q : I \to \mathbb{R}$ be continuous. An integrating factor of the linear first-order equation $y' + p(x) y = q(x)$ is a differentiable function $\mu : I \to \mathbb{R}$ with $\mu(x) \ne 0$ for each $x \in I$ and $$\mu'(x) = p(x)\, \mu(x) \quad \text{for each } x \in I.$$ For such $\mu$ and each differentiable $y$ on $I$, $\mu \left(y' + p y\right) = (\mu y)'$.

Remarks

If $P$ is an antiderivative of $p$ on $I$, then $\mu(x) = e^{P(x)}$ is an integrating factor. Each integrating factor is a nonzero constant times this one. After multiplication by $\mu$, the equation becomes $(\mu y)' = \mu q$. One integration gives the solution formula. More generally, an integrating factor is a function that makes an equation exact after multiplication. See integrating factor for a non-exact equation.

Explanations

L1 human

The left side $y' + p y$ is not the derivative of a simple expression. After multiplication by the right function $\mu$, it becomes the derivative of $\mu y$, by the product rule. Then you can integrate both sides directly. For $y' + y = x$, the factor is $e^x$, and the equation becomes $(e^x y)' = x e^x$.