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ODE, Homework 3
Due Thursday, June 26, 2008
1.
(6.3.31) We saw in class that if
f
is periodic with period
T
(ie
f
(
t
+
T
) =
f
(
t
) for all
t
and some ﬁxed minimal positive number
T
) then
L{
f
(
t
)
}
=
R
T
0
e

st
f
(
t
)
dt
1

e

sT
.
Use this to compute the Laplace transform of the “sawtooth wave”
f
(
t
) =
t
for 0
≤
t <
1
and
f
(
t
+ 1) =
f
(
t
) for all
t
∈
R
.
2.
(6.4.19 slightly modiﬁed) Consider the initial value problem
y
00
+
y
=
f
(
t
),
y
(0) = 0,
y
0
(0) = 0, where
f
(
t
) =
u
0
(
t
) + 2
∑
n
k
=1
(

1)
k
u
kπ
(
t
).
(a) Draw the graph of
f
(
t
) on an the interval 0
≤
t
≤
6
π
.
(b) Use the Laplace transform to solve the initial value problem.
(c) Let
n
= 3 and plot the graph of the solution for 0
≤
t
≤
6
π
.
(d) Describe qualitatively what happens to the solutions as
n
→ ∞
. One or two sentences
should suﬃce.
3.
(6.5.18 slightly modiﬁed) Consider the initial value problem
y
00
+
y
=
g
(
t
),
y
(0) = 0,
y
0
(0) = 0, where
f
(
t
) =
u
0
(
t
) + 2
∑
n
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 Spring '11
 Peters
 Differential Equations, Equations

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