A separable 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 formula_1-separation of variables which is accomplished by writing the solution as a particular fixed function of the coordinates multiplied by a product of functions of each individual coordinate. Laplace's equation on formula_2 is an example of a partial differential equation that admits solutions through formula_1-separation of variables; in the three-dimensional case this uses 6-sphere coordinates.
Example.
For example, consider the time-independent Schrödinger equation
formula_4
for the function formula_5 (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function formula_6 in three dimensions is of the form
formula_7
then it turns out that the problem can be separated into three one-dimensional ODEs for functions formula_8, formula_9, and formula_10, and the final solution can be written as formula_11. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.)