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15.1 Eigenfunction Expansions 513

x
–0.5 0 0.5 1 1.5 2
0

–1

–2

–3

–4
FIGURE 15.1 Eigenfunction expansion in Example 15.3.
Figure 15.1 shows a graph of f compared with the fifteenth partial sum of this series. The
graph is drawn on a slightly larger interval than [0,π/2]. This eigenfunction expansion appears to
converge rapidly to f (x) on [0,π/2], but the graphs diverge from each other outside this interval.
The eigenfunction expansion is unrelated to f (x) outside of the interval of the Sturm-Liouville
problem.

The following example illustrates an eigenfunction expansion in which the eigenfunctions
are not just sines and cosines.

EXAMPLE 15.4
We will expand f (x)= x in a series of the eigenfunctions of the regular Sturm-Liouville problem

y + 4y + (3 + λ)y = 0; y(0) = y(1) = 0
on [0,1]. First we must generate the eigenvalues and eigenfunctions. Put y = e rx to obtain the
characteristic equation of the differential equation:
2
r + 4r + (3 + λ) = 0,
with roots
√
r =−2 ± 1 − λ.
Take cases on λ.
Case 1: λ = 1
Then
y(x) = c 1 e −2x + c 2 xe −2x .
Now
y(0) = c 1 = 0 and y(1) = c 2 e −2 = 0
so c 2 = 0 and y is the trivial solution. Therefore, 1 is not an eigenvalue of this problem.

Case 2: 1 − λ> 0
2
Write 1 − λ = α with α> 0 to obtain
y(x) = c 1 e (−2+α)x + c 2 e (−2−α)x .

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October 14, 2010 15:20 THM/NEIL Page-513 27410_15_ch15_p505-562