Separation of variables
Separation of variables is a classical technique to construct special solutions u to an ODE by using the ansatz for functions f and g of space and time separately. Using the fact that a function of space and a function of time can only be equal if both are constant, one can then decouple the PDE into an ODE in time and a PDE in space, which are presumably easier to solve.
In the case of linear time-translation-invariant equations, one can often express a general solution (under suitable regularity conditions) as a superposition of solutions constructed via separation of variables, for instance by using spectral theory. However this phenomenon is generally unavailable in non-linear or time-dependent contexts.