Given a boundary value problem with independent variables $x_1,x_2, \dots , x_n$ and a PDE say $U(x_i, y, \partial_j y,\partial_{ij} y, \dots )=0$ we typically begin constructing a general solution by making the ansatz $y = F_1(x_1)F_2(x_2) \cdots F_n(x_n)$ where each $F_i$ is a function of just one variable. Next, we plug this product solution into the given PDE and typically we obtain a family of ODEs for $F_1, F_2, \dots, F_n$ which are necessarily related by a characteristic constant. Often, boundary values are given which force a particular spectrum of possible constants. Each allowed value gives us a solution and the general solution is assembled by summing the possible BV solutions. (there is more for nonhomogenous problems etc... here I sketch the basic technique I learned in second semester DEqns as a physics undergraduate)

Examples, the heat equation, the wave equation, Laplace's equation for the electrostatic case, Laplace's equation as seen from fluids, Schrodinger's equation. With the exception of the last, these are not quantum mechanical. My question is simply this:

what is the physical motivation for proposing a product solution to the classical PDEs of mathematical physics?

I would ask this in the MSE, but my question here is truly physical. Surely the reason "because it works" is a reason, but, I also hope there is a better reason. At the moment, I only have some fuzzy quantum mechanical reason and I have to think there must be a classical physical motivation as well since these problems are not found in quantum mechanics.

Thanks in advance for any insights !

This post imported from StackExchange Physics at 2014-06-06 20:04 (UCT), posted by SE-user James S. Cook