---
tag: 47NFRVB564
kind: concept
name: Separation of variables
msc: [34A05, 35C05]
sources:
  - url: https://en.wikipedia.org/wiki/Separation_of_variables
same_as:
  wikidata: Q913323
  wikipedia: Separation of variables
---

## Summary

Separation of variables is a method for solving differential equations by splitting the dependence on different variables.
For an ordinary differential equation $y' = g(x)\, h(y)$, one moves the $y$ terms to one side and the $x$ terms to the other and integrates.
Such equations are called [[separable equations|074E9A8ZJK]].
For a linear partial differential equation, one looks for solutions that are products of functions of one variable each.
This turns the equation into several ordinary differential equations.
It is one of the oldest and most used solution methods.
See the [[definition|PR2DV03M4H]].

## Explanations

### L1 human

The idea is to put everything about one variable on one side of the equation and everything about the other variable on the other side.
For $y' = x y$, one writes $dy / y = x\, dx$, integrates both sides to get $\ln\abs{y} = x^2/2 + C$, and finds $y = A e^{x^2/2}$.
For the heat equation $u_t = u_{xx}$, one guesses $u(x, t) = X(x)\, T(t)$, and the equation splits into one equation for $X$ and one for $T$.
Sums of such product solutions then give many more solutions, which leads to Fourier series.