, what is the formula for
c
n
?
Or simpler:
What is a regular
SturmLiouville problem?
(c) Fillers. Short questions that take about a minute or less to answer correctly, if you know the answer.
Example: If
x
3
=
∞
∑
n
=1
b
n
sin
nπx
2
for 0
< x <
2; what does
∞
∑
n
=1
b
n
sin
7
nπ
2
equal?
The material of the course can be divided into the following topics, which correspond to Chapters 2, 3, 5, 6, 7,
8 of the textbook:
I. Fourier series.
Things to know: What is a Fourier series, how does one compute the coeﬃcients, convergence
facts.
Typical problem:
A relatively simple function is given, say
f
(
x
) =
x
2
in the interval [

1
,
2], or a
functions with a few more jumps in some other interval, and you may be asked to expand it into a Fourier
series, asked questions about its convergence.
Similarly for sine and cosine series.
II.
SturmLiouville Problems.
Know the difference between regular, singular and periodic.
Main properties of
eigenvalues and eigenfunctions. Rayleigh quotient.
Here are some problems. Exercises 4, 5 are good practice exercises, but one could argue that it might not be
fair to have similar ones in the final. One could argue.
1. Consider the problem
y
′′

xy
′
+
λ
(
x
2
+ 1)
y
= 0
,
0
< x <
1
,
y
(0) = 0
,
y
′
(1) = 0
.
Show that it is a regular SturmLiouville problem. Be sure to state clearly what
p
(
x
)
, q
(
x
), and
σ
(
x
)
are.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
2. Consider the SturmLiouville problem
y
′′
+ 4
y
′
+ 5
y
+
λy
= 0
,
0
≤
x
≤
2
π,
y
(0) = 0
, y
(2
π
) = 0
.
(a) Find all eigenvalues and corresponding eigenfunctions.
(b) Suppose the function
f
(
x
) =
{
x,
0
< x < π,
0
,
π
≤
x <
2
π,
is expanded into a series of eigenfunctions. Write
out a formula for the coeﬃcients.
Do not compute any integrals, except if you have time
to spare.
3. Compute all eigenfunctions and eigenvalues of the following regular SturmLiouville problem.
y
′′
+
λy
= 0
,
0
< x <
1
,
y
′
(0)

y
(0) = 0
,
y
(1) = 0
.
4. Compute all eigenfunctions and eigenvalues of the following regular SturmLiouville problem.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '13
 Schonbek
 Boundary value problem, Partial differential equation, regular SturmLiouville problem

Click to edit the document details