Aseparable partial differential equation can be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation of variables. It generally relies upon the problem having some special form or symmetry. In this way, the partial differential equation (PDE) can be solved by solving a set of simpler PDEs, or even ordinary differential equations (ODEs) if the problem can be broken down into one-dimensional equations.
The most common form of separation of variables is simple separation of variables. A solution is obtained by assuming a solution of the form given by a product of functions of each individual coordinate. There is a special form of separation of variables called 
(This should not be confused with the case of a separable ODE, which refers to a somewhat different class of problems that can be broken into a pair of integrals; see separation of variables.)
For example, consider the time-independent Schrödinger equation
![{\displaystyle [-\nabla ^{2}+V(\mathbf {x} )]\psi (\mathbf {x} )=E\psi (\mathbf {x} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab61a33bb9e3ffad587e95036e68561f1d9884e)
for the function 
then it turns out that the problem can be separated into three one-dimensional ODEs for functions 
