I am studying Laplace's equation in my electrodynamics course (using griffiths intro to electrodynamics). I am watching a youtube video stepping through the separation of variables method for solving the PDE. It seems to be a common PDE that comes up repeatedly in physics (Helmholtz eqn, Poisson eqn) and I thought it would be worthwhile to understand where the solution comes from. In the text Griffiths also touches on two additional points: - "A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere" - A uniqueness theorem A first question that came to my mind: why do we assume that separation of variables works? How can we assume that the solution is of the form [itex]f(x)g(y)[/itex]? What if it had an expression like [itex]x^y[/itex]?