what is separation of variables

Solution 1:

Some functions (not all) $\psi (x,t)$ can be written as a product of a function of $x$ and another function of $t$. For example, $\psi (x,t)=xt$ can be, while $\psi_2 (x,t)=x^2+t^2$ cannot. The author is guessing that this will yield a solution to the problem and will go on to show that it does. After some manipulation the equation comes to something like $f(x)=g(t)$ where the left does not depend on $t$ and the right does not depend on $x$. Then you argue that since the left does not depend on $t$, the right really doesn't either, and both sides must equal some constant. So now you are solving $f(x)=g(t)=c$. As equations in single variables it is usually easier. Proving that this yields a solution is easy. Proving that all solutions come as a linear combination of solutions of this form is harder.

Solution 2:

Regarding your question about the generality of separation of variables, there is an extremely beautiful Lie-theoretic approach to symmetry, separation of variables and special functions, e.g. see Willard Miller's book [1]. I quote from his introduction:

This book is concerned with the relationship between symmetries of a linear second-order partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner. It is an introduction intended for anyone with experience in partial differential equations, special functions, or Lie group theory, such as group theorists, applied mathematicians, theoretical physicists and chemists, and electrical engineers. We will exhibit some modern group-theoretic twists in the ancient method of separation of variables that can be used to provide a foundation for much of special function theory. In particular, we will show explicitly that all special functions that arise via separation of variables in the equations of mathematical physics can be studied using group theory. These include the functions of Lame, Ince, Mathieu, and others, as well as those of hypergeometric type.

This is a very critical time in the history of group-theoretic methods in special function theory. The basic relations between Lie groups, special functions, and the method of separation of variables have recently been clarified. One can now construct a group-theoretic machine that, when applied to a given differential equation of mathematical physics, describes in a rational manner the possible coordinate systems in which the equation admits solutions via separation of variables and the various expansion theorems relating the separable (special function) solutions in distinct coordinate systems. Indeed for the most important linear equations, the separated solutions are characterized as common eigenfunctions of sets of second-order commuting elements in the universal enveloping algebra of the Lie symmetry algebra corresponding to the equation. The problem of expanding one set of separable solutions in terms of another reduces to a problem in the representation theory of the Lie symmetry algebra.

For an example of effective Lie-theoretic algorithms for first-order ODEs see Bruce Char's paper[2], from which the following useful tables are extracted.

1 Willard Miller. Symmetry and Separation of Variables.
Addison-Wesley, Reading, Massachusetts, 1977 (out of print)
http://www.ima.umn.edu/~miller/separationofvariables.html
http://gigapedia.com/items:links?id=64401

2 Bruce Char. Using Lie transformation groups to find closed form solutions to first order ordinary differential equations.
SYMSAC '81. Proceedings of the fourth ACM symposium on Symbolic and algebraic computation.
http://portal.acm.org/citation.cfm?id=806370