Separable differential equation

One where the $x$ 's and $y$ 's can be sorted onto opposite sides, then each side integrated separately.

A separable differential equation can be rearranged into the form $g(y)\,dy=f(x)\,dx$ — every $y$ (including $dy$ ) on one side, every $x$ (including $dx$ ) on the other. Integrating each side separately, then solving for $y$ if possible, gives the general solution. This is the most common technique for a first-order equation that's solvable in closed form at all.

Examples

$y'=xy$ separates to $\frac{dy}{y}=x\,dx$ , integrating to $\ln|y|=\frac{x^2}{2}+C$ , so $y=Ae^{x^2/2}$ .