(a)
(10 points)
Express
f
as a Fourier series in the interval [
−
1
,
1]. Leave the coeﬃcients indicated, do
NOT
evaluate any integrals. But
anything that is 0 should not appear in the final expression.
Fancy ways of writing 0 will be penalized.
Solution.
Because the function is even, only cosine terms will be present. Thus
f
(
x
) =
1
2
a
0
+
∞
∑
n
=1
a
n
cos
nπx,
where
a
n
=
∫
1
−
1
x
2
cos
nπx dx.
(b)
(5 points)
At what points
x
of the interval [
−
1
,
1] is
f
(
x
) equal to the sum of its Fourier series?
Solution.
Because
f
is continuous at all points of the interval, and because
f
(1
−
) =
f
(1+), the series
converges to
f
(
x
) at
ALL
points of [
−
1
,
1].
(c)
(5 points)
To what values, if any, does the Fourier series converge at the points
x
= 3
.
2
,
4
.
9, and
−
6?
Solution.
The function that is the sum of the Fourier series is periodic of period 2 and coincides with
x
2
for

x
 ≤
1. Thus:
•
Limit or sum at 3
.
2 = same as at
−
0
.
8, so to (
−
0
,
8)
2
= 0
.
64.
•
Limit or sum at 4
.
9 = same as at 0
.
9, so to 0
.
9
2
= 0
.
81.
•
Limit or sum at
−
6 = same as at 0, so to 0
2
= 0.
2. Consider the following problem:
(1 +
x
2
)
y
′′
(
x
) + 2
xy
′
+
λ
(1 +
x
2
)
y
=
0
,
0
< x <
1
,
y
′
(0) =
y
′
(1) = 0
.
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2
(a)
(10 points)
Write it out in the form of a regular Sturm–Liouville problem. What are
p, q, σ
?
Solution.
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 Spring '13
 Schonbek
 Sin, Boundary value problem, Partial differential equation, Sturm–Liouville theory

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