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A separable partial differential equation (PDE) is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This generally relies upon the problem having some special form or symmetry. In this way, the 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 in which 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 $R$-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 ${\mathbb{ℝ}}^n$ is an example of a partial differential equation which admits solutions through $R$-separation of variables.

(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.)

Example

For example, consider the time-independent Schrödinger equation

$[-{\mathrm{\nabla }}^2+V(\mathbf{x})]\psi (\mathbf{x})=E\psi (\mathbf{x})$

for the function $\psi (\mathbf{x})$ (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function $V(\mathbf{x})$ in three dimensions is of the form

$V(x_1,x_2,x_3)=V_1(x_1)+V_2(x_2)+V_3(x_3),$

then it turns out that the problem can be separated into three one-dimensional ODEs for functions ${\psi }_1(x_1)$, ${\psi }_2(x_2)$, and ${\psi }_3(x_3)$, and the final solution can be written as $\psi (\mathbf{x})={\psi }_1(x_1)\cdot {\psi }_2(x_2)\cdot {\psi }_3(x_3)$. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.)