What is. An integrating factor

For a first-order linear differential equation in the standard form:

$$\frac{dy}{dx} + P(x)y = Q(x)$$

The integrating factor, denoted by $\mu(x)$, is given by the formula:

$$\mu(x) = e^{\int P(x)\,dx}$$

Multiplying the entire differential equation by this integrating factor allows the left side to be expressed as the derivative of a product, specifically $\frac{d}{dx}(\mu(x)y)$, which can then be integrated to find the solution for $y$.