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CDS 140a Final Examination
J. Marsden, December, 2009
The Exam is due by noon on Friday, December 11, 2009
Attempt SEVEN of the following ten questions.
Each question is worth 20 points.
The exam time limit is three hours; no aids of any kind
(including the internet, computers, books, calculators).
Print Your Name
:
The SEVEN questions to be graded
:
1
2
3
4
5
6
7
8
9
10
/140
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1. Consider the system
˙
x
=
y
˙
y
=
αx

βy

x
3

γx
2
y
3
where
α >
0,
β
≥
0 and
γ
≥
0 are constants.
(a) If
β
= 0 and
γ
= 0 compute a conserved energy
E
(
x,y
) for the system.
(b) Compute the time derivative of the energy
E
(
x,y
) for any
α,β,γ
. Do
solutions exist for all time?
(c) Show that for any
α,β,γ
, if (
x
(
t
)
,y
(
t
)) is a solution, so is (

x
(
t
)
,

y
(
t
)).
Is the system timereversible? What does this say about the phase por
trait?
2. (Continuation of Problem 1).
(a) Find all the ﬁxed points of the system and determine their stability in
terms of
α
,
β
and
γ
.
(b) Using
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 Fall '09
 Marsden

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