
6.5. Equations Containing Arbitrary Functions 627
where the λ
n
and y
n
(x) are the eigenvalues and corresponding eigenfunctions of the Sturm–
Liouville problem for the following second-order linear ordinary differential equation with
homogeneous boundary conditions:
[p(x)y
′
x
]
′
x
+ [λ − q(x)]y = 0,
s
1
y
′
x
+ k
1
y = 0 at x = x
1
,
s
2
y
′
x
+ k
2
y = 0 at x = x
2
.
(5)
The functions U
n
= U
n
(t, τ) are determined by solving the Cauchy problem for the linear
ordinary differential equation
U
′′
n
+ a(t)U
′
n
+ λ
n
b(t)U
n
= 0,
U
n
t=τ
= 0, U
′
n
t=τ
= 1.
(6)
The prime denotes the derivative with respect to t, and τ is a free parameter occurring in
the initial conditions.
The functions Λ
1
(x, t)