TITLE: How does one motivates the method of separation of variables when teaching PDE's?
QUESTION [15 upvotes]: I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math behind it? It's not just because the guy who fancied it is very smart, right? (Although I feel like it does give students this impression...)
(Background: At Berkeley there is a course called Math 54, in which students learn linear algebra, linear ODEs and then 1 or 2 weeks of PDEs. Teaching Separation of variables is always my nightmare...)

REPLY [2 votes]: At the level of your Math 54 I suggest the following: By the time you
get to a heat or wave equation, the students are used to seeing functions
such as $ce^{at}$, $c_1\cos(at)+c_2\sin(at)$, as solutions to simple ODE. It is
somewhat natural then to let the constants $c$ become functions of $x$.
It becomes a reasonable thing to try, an extension of ``undetermined coefficients''.
 (The Notes on Differential Equations
on my web page uses this description.)