Several of the common partial differential equations encountered in physics are mentioned as easy to solve using the variable separable method. However I do not understand how does one guarantee that the solutions generated using this method is the complete set of solutions. I am finding it quite hard to believe that such a method can produce the entire set of solutions.

Can anyone explain if it indeed does generate the complete set of solutions to a PDE. If not, a counterexample would work. Also, I would like to know if we use this because it is consistent and easy to work with?