First order differential equations

First-order equations: integrating factors

An equation $\frac{dy}{dx}+P(x)y=Q(x)$ can't usually be separated. Multiply through by the integrating factor $I=e^{\int P\,dx}$: the left side becomes $\frac{d}{dx}(Iy)$, exactly the derivative of a product, so one integration solves it.