Separation of variables for partial differential equations What class of Partial Differential Equations can be solved using the method of separation of variables?
 A: There is an extremely beautiful Lie-theoretic approach to separation of variables,
e.g. see Willard Miller's book [1] (freely downloadable). 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 modem 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 Lam6, 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.

[1] Willard Miller. Symmetry and Separation of Variables.
Addison-Wesley, Reading, Massachusetts, 1977 (out of print)
A: For example the linear homogeneous PDEs with dependent variable $u$ and independent variables $x$ and $y$ , the separable condition is that the PDEs can rewrite to the form $\dfrac{\sum\limits_{a_1=0}^{b_1}M_{a_1}(x)X^{[a_1]}(x)}{\sum\limits_{a_2=0}^{b_2}N_{a_2}(x)X^{[a_2]}(x)}=\dfrac{\sum\limits_{a_3=0}^{b_3}P_{a_3}(y)Y^{[a_3]}(y)}{\sum\limits_{a_4=0}^{b_4}Q_{a_4}(y)Y^{[a_4]}(y)}$ when letting $u(x,y)=X(x)Y(y)$ .
For example, the PDE $x^2u_{xy}-yu_{yy}+u_x-4u=0$ mentioned in The canonical form of a nonlinear second order PDE is an unseparable example while the PDE $u_{xy}-yu_{yy}+u_x-4u=0$ is a separable example.
Start from the PDEs with three independent variables, the separable conditions are more difficult to described, since for example the linear homogeneous PDEs with dependent variable $u$ and independent variables $x$ , $y$ and $z$ , the PDEs are separable when the PDEs not only can rewrite to the form $\dfrac{\sum\limits_{a_1=0}^{b_1}M_{1,a_1}(x)X^{[a_1]}(x)}{\sum\limits_{a_2=0}^{b_2}M_{2,a_2}(x)X^{[a_2]}(x)}+\dfrac{\sum\limits_{a_3=0}^{b_3}M_{3,a_3}(y)Y^{[a_3]}(y)}{\sum\limits_{a_4=0}^{b_4}M_{4,a_4}(y)Y^{[a_4]}(y)}+\dfrac{\sum\limits_{a_5=0}^{b_5}M_{5,a_5}(z)Z^{[a_5]}(z)}{\sum\limits_{a_6=0}^{b_6}M_{6,a_6}(z)Z^{[a_6]}(z)}=0$ when letting $u(x,y,z)=X(x)Y(y)Z(z)$ , but also when the PDEs can rewrite to the form $\dfrac{\sum\limits_{a_1=0}^{b_1}M_{1,a_1}(x)X^{[a_1]}(x)}{\sum\limits_{a_2=0}^{b_2}M_{2,a_2}(x)X^{[a_2]}(x)}+\dfrac{\sum\limits_{a_3=0}^{b_3}M_{3,a_3}(y)Y^{[a_3]}(y)}{\sum\limits_{a_4=0}^{b_4}M_{4,a_4}(y)Y^{[a_4]}(y)}+\dfrac{\sum\limits_{a_3=0}^{b_3}N_{3,a_3}(y)Y^{[a_3]}(y)\sum\limits_{a_5=0}^{b_5}M_{5,a_5}(z)Z^{[a_5]}(z)}{\sum\limits_{a_4=0}^{b_4}N_{4,a_4}(y)Y^{[a_4]}(y)\sum\limits_{a_6=0}^{b_6}M_{6,a_6}(z)Z^{[a_6]}(z)}=0$ when letting $u(x,y,z)=X(x)Y(y)Z(z)$ .
