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508 CHAPTER 15 Special Functions and Eigenfunction Expansions

Case 2: λ< 0
2
kx
Say λ =−k .Now y(x) = c 1 e + c 2 e −kx . The condition y(L) = y(−L) gives us
kL
kL
c 1 e −kL + c 2 e = c 1 e + c 2 e −kL .
Write this as
kL
kL
c 1 (e −kL − e ) = c 2 (e −kL − e ).
kx

This implies that c 1 = c 2 ,so y(x) = c 1 (e + e −kx ).Now y (−L) = y (L) gives us
kL
kL
c 1 k(e −kL − e ) = c 1 k(e − e −kL ).
This implies that c 1 =−c 1 , hence c 1 =0, so c 2 =0 also. This problem has only the trivial solution,
so there is no negative eigenvalue.
Case 3: λ> 0

2
2
Say λ = k . The general solution of y + k y = 0is

y(x) = c 1 cos(kx) + c 2 sin(kx).
Now
y(−L) = c 1 cos(kL) − c 2 sin(kL) = y(L) = c 1 cos(kL) + c 2 sin(kL).
This implies that c 2 sin(kL) = 0. Next,

y (−L) = kc 1 sin(kL) + kc 2 cos(kL) = y (L) =−kc 1 sin(kL) + kc 2 cos(kL),
implying that kc 1 sin(kL) = 0. If sin(kL) = 0, then c 1 = c 2 = 0 and we have only the trivial
solution. For a nontrivial solution, we must have sin(kL) = 0, so at least one of the constants c 1
and c 2 can be chosen nonzero. But sin(kL) = 0 is satisfied if kL = nπ, with n a positive integer
(positive because we chose k > 0). Since λ = k , the eigenvalues of this problem, indexed by n,
2
are
nπ
2
λ n = for n = 1,2,···
L
and corresponding eigenfunctions are
nπx nπx

y n (x) = c 1 cos + c 2 sin ,
L L
with c 1 and c 2 any constants, not both zero.
By choosing n = 0 and c 2 = 0but c 1 = 0, we obtain a constant eigenfunction corresponding
to the eigenvalue 0. This consolidates cases 1 and 3.

Bessel’s equation will provide an example of a singular Sturm-Liouville problem (here we
will have r(0) = 0).
We may also have a Sturm-Liouville problem in which r(a) = r(b) = 0, but there are no
boundary conditions specified. In this event we impose the condition that eigenfunctions must
be bounded on [a,b]. The Legendre differential equation will provide an example of this type of
problem.
A Fourier sine series on [0, L] is an expansion in the eigenfunctions of Example 15.1. A
Fourier series on [−L, L] is an expansion in the eigenfunctions of Example 15.2. This raises an
intriguing question. If a Sturm-Liouville problem on [a,b] has eigenfunctions ϕ 1 (x),ϕ 2 (x),···,

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October 14, 2010 15:20 THM/NEIL Page-508 27410_15_ch15_p505-562