So we just started finding general solutions for homogenous&linear two-variabled PDEs by using separation of variables in my engineering-math class. There the professor tells us to assume the solution of a PDE is in the form of F(x)*G(t). But when is the solution in the form of F(x)*G(t)? When does separation of variables work and not (does it always work, in theory atleast, on linear PDEs?)? For multivariabled linear PDEs will the solution be on the form of F(x)*G(y)*H(z)*J(t)... etc? I'm a bit confused currently... all help is appreciated.