Separation of variables

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.

Suppose a differential equation can be written in the form

which we can write more simply by letting ${\displaystyle y=f(x)}$:

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

so that the two variables x and y have been separated. 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.

Some 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,

because of the substitution rule for integrals.

If one can evaluate the two integrals, one can find a solution to the differential equation. Observe that this process effectively allows us to treat the derivative ${\displaystyle {\frac {dy}{dx}}}$ as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below.

...
Wikipedia