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Problem Set 5
February 16, 2007
Due February 23, 2007
ACM 95b/100b
3pm in Firestone 303
Niles A. Pierce
(2 pts) Include grading section number
1. (3
×
8 pts) Solve each of the following boundary value problems
a)
y
00
+
y
= 0
,
y
(0) = 0
,
y
0
(
π
) = 1
b)
y
00
+
y
= 0
,
y
0
(0) = 1
, y
(
L
) = 0
c)
y
00
+ 4
y
= sin
x,
y
(0) = 0
,
y
(
π
) = 0
2. (15 pts) Consider the eigenvalue problem
y
00
+
λy
= 0
,
y
(0) = 0
,
y
(
L
) = 0
.
Show that there are no complex eigenvalues.
3. (15 pts) Find the Fourier series for the function
f
(
x
) =
H
(
x
) on the interval
x
∈
[

π,π
],
where
H
(
x
) is the Heavyside step function. Plot a sequence of partial sums and observe that the
maximum overshoot does not tend to zero as the number of terms is increased. This behavior
is known as Gibb’s phenomenon. Use your plots to estimate the approximate magnitude of the
maximum overshoot as the number of terms increases to inﬁnity. What happens to the location of
the maximum overshoot as the number of terms increases?
4. (3
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 Winter '09
 NilesA.Pierce

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