Integrating factor

A multiplier $\mu(x)=e^{\int P(x)\,dx}$ that turns a linear DE's left side into a product-rule derivative.

An integrating factor $\mu(x)=e^{\int P(x)\,dx}$ solves a first-order LINEAR differential equation $y'+P(x)y=Q(x)$ by multiplying both sides by $\mu(x)$ , which makes the left side collapse into the derivative of a single product: $\frac{d}{dx}[\mu(x)y]=\mu(x)Q(x)$ . Integrating both sides then solves directly for $y$ , even when the equation isn't separable.

Examples

For $y'+2y=e^x$ , $\mu(x)=e^{\int2\,dx}=e^{2x}$ , turning the equation into $\frac{d}{dx}[e^{2x}y]=e^{3x}$ .