separation ansatz

A trial solution u(x,t) = X(x)T(t) -- a product of a function of x alone and a function of t alone -- substituted into a PDE to split it into two ordinary differential equations.

The separation ansatz (the method of separation of variables) seeks a solution of the form $u(x,t)=X(x)T(t)$ ; substituting into a linear homogeneous PDE and dividing by $X(x)T(t)$ isolates every $x$ -dependent piece on one side of the equation and every $t$ -dependent piece on the other, forcing both sides to equal a shared constant and splitting the single PDE into two ordinary differential equations, one in $x$ and one in $t$ .