ABSTRACT
In addition to the techniques described in Chapter 10, the method of separation of variables is a powerful tool with which to solve linear partial differential equations. To begin with, we seek solutions of (11.1.2) in the form
u(x, t) = X(x)T (t). (11.1.3)
Substitution of this into (11.1.2) leads to the identity
X dT
dt = T
d2X
dx2
or 1
T
dT
dt =
X
d2X
dx2 . (11.1.4)
Now 1T dT dt is a function of t only, while
d2X dx2 is a function of x only. Con-
sequently, both sides of (11.1.4) must be equal to a constant, say λ. Thus X and T must satisfy the ordinary differential equations
dT
dt − λT = 0,
d2X
dx2 − λX = 0. (11.1.5)
X(x) = exp± √ λx, T (t) = expλt.
