Chapter 4
Thus (4.9) becomes
(
θ
0
)
2
2
−
cos
θ
=1
.
In particular, at the initial moment,
t
=0
,
[
θ
0
(0)]
2
2
−
cos[
θ
(0)] = 1
.
Since
θ
(0) = 0, we get
[
θ
0
(0)]
2
2
−
cos 0 = 1
⇒
[
θ
0
(0)]
2
=4
⇒
θ
0
(0) = 2
or
θ
0
(0) =
−
2
.
11.
The “damping coeﬃcient” in the Rayleigh equation is
b
=(
y
0
)
2
−
1. Thus, for low velocities
y
0
,w
ehav
e
b<
0, and
b>
0 for high velocities. Therefore, the low velocities are boosted,
while high velocities are slowed, and so one should expect a limit cycle.
13.
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, low velocities

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