A linear homogeneous PDE takes the form
where L is a linear operator. There is also the associated inhomogeneous PDE
where the forcing term F is given.
Linear equations obey the principle of superposition, which allows one to synthesize general solutions from specific ones (such as those obtained from separation of variables). Two other very useful tools related to the principle of superposition are Duhamel's formula and the fundamental solution.
When the linear equation is constant-coefficient then the Fourier transform is an excellent tool for solving these equations. For variable-coefficient linear operators, then generalizations of the Fourier transform (such as scattering transforms or Fourier integral operators) can be employed.