14.3 Separation of variables 393
The partial derivatives are
Therefore,
as required. The function can be written as
f(x, 1 t) 1 = 1 (x 1 + 1 vt)
21 + 1 2(x 1 − 1 vt)
21 = 1 F(x 1 + 1 vt) 1 + 1 G(x 1 − 1 vt)
0 Exercises 1, 2
EXAMPLE 14.2Verify that the function
f(x, 1 t) 1 = 1 a exp[−b(x 1 − 1 vt)
2]
is a solution of the wave equation (14.1).
We have
Therefore.
0 Exercise 3
14.3 Separation of variables
When a partial differential equation, involving two or more independent variables,
can be reduced to a set of ordinary differential equations, one for each variable, the
equation is calledseparable. The solutions of the partial differential equation are then
products of the solutions of the ordinary equations. All the examples discussed in this
chapter are of this type.
We demonstrate the essential principles of the method of separation of variables
by considering the simplest first-order equation in two variables
∂
∂
=
∂
∂
22222f 1
x
f
v t
∂
∂
=−+−
−−
2222 2212
f
t
abvv vb x t()exp()b x t
∂
∂
=−+ −
−−
2222212
f
x
ab b x t()exp()vvb x t
6
1
22222=
∂
∂
=
∂
∂
f
x
f
v t
∂
∂
=−,
∂
∂
=;
∂
∂
=− + ,
∂
∂
=
f
x
xt
f
x
f
t
xt
f
t
62 6 2 6
22222vvv 66
2v