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MATH 251
Fall 2003
Final Exam
Decemebr 15, 2003
ANSWERS:
1.
B
2.
B or C
3.
C
4.
A
5.
B
6.
A
7.
(a) Critical points are
(1
,
1)
and
(1
,

1)
(b)
(1
,
1)
is unstable (a saddle point);
(1
,

1)
is unstable (a spiral point/spiral source)
8.
y
(
t
) =

7
6
e

3
t
+
1
2
e
t

1
3
t
+
2
3
9.
m
= 2
,
γ
= 6
,
k
=
mg
L
= 4
(a)
2
u
” + 6
u
0
+ 4
u
= 0
or
u
” + 3
u
0
+ 2
u
= 0
,
u
(0) = 0
,
u
0
(0) = 5
Solution:
u
(
t
) = 5
e

t

5
e

2
t
(b) (Trick question!) The system is overdamped
. There is no oscillation, therefore, there is no quasi
period. Also acceptable: quasiperiod =
∞
.
10.
(a)
y
=

3
,
0
,
3
(b)
y
=

3
is asymptotically stable;
y
= 0
is unstable;
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This note was uploaded on 07/23/2008 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.
 Spring '08
 CHEZHONGYUAN
 Differential Equations, Equations, Critical Point, Partial Differential Equations

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