In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

A differential equation for the unknown ${\displaystyle f(x)}$ is separable if it can be written in the form

where ${\displaystyle g}$ and ${\displaystyle h}$ are given functions. This is perhaps more transparent when written using ${\displaystyle y=f(x)}$ as:

So now as long as h(y) ≠ 0, we can rearrange terms to obtain:

where the two variables x and y have been separated. Note dx (and dy) can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of dx as a differential (infinitesimal) is somewhat advanced.

Those who dislike Leibniz's notation may prefer to write this as

but that fails to make it quite as obvious why this is called "separation of variables". Integrating both sides of the equation with respect to ${\displaystyle x}$, we have

or equivalently,