Separable Equations

A separable differential equation is a first order differential equation that can be written in the form

$$\frac{dy}{dx} = \frac{f(x)}{h(y)}.$$

These can be solved by integration, by noting that

$$h(y) dy = f(x) dx,$$

hence

$$\int h(y) dy = \int f(x) dx.$$

This latter equation defines $y$ implicitly as a function of $x$, and in some cases it is possible to explicitly solve for $y$ as a function of $x$.