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 $I$, $\mu \left(y' + p y\right) = (\mu y)'$.