Math 2280 Practice Exam 2
-
University of Utah
Spring 2013
Name:
5
-L
i
This is a 50 minute exam. Please show all your work, as a worked
problem is required for full points, and partial credit may be rewarded for
some work in the right direction.
1
1. (25 points) Population Models
For the population model
=3P(P-5)
what are the equilibrium populations, and for each equilibrium poplu
ation determine if it is a stable or unstable. Draw the corresponding
phase diagram. Then, use separation of variables to solve the popu
lation model exactly with the initial population P(O) = 2.
Oo7f=5
Pc
o(P
-
io}
*
-
3
2
More room for problem 1, if you need it.
p9
Ce
r
I5
p-9
7
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5
—
9
5
Ce’
I-c
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9Z-7(
5
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IC)
L4
JV i;
Ii
=
3
C
as
2. (20 points) Mechanical Systems
C
k
x
1 above, find the equa
For the mass-spring-dashpot system drawn
tion that describes its motion with the parameters:
Tn
=
3;
c
=
30;
k
=
63;
and initial conditions:
xo
=
2;
0
V
2.
Is the system overdamed, underdamped, or critically damped?
1
4
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N
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t
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—C
0
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r
+
ON
÷
CD
CD
0
CD
0
0
0
0
0
(D
3. (20 points) Higher Order Linear Homogeneous Differential Equations with
Constant Coefficients
Find the general solution to the differential equation:
3 + y”
+ 2y
Hint: The polynomial
multiplicity 2.
+
—
2
+x
12y’ + 8
—
—
0.
12r + 8 has x
—
1 as a root of
(kiacYi)
L
1
L
Crl) (r± +)
-
yeYce
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6
4. (15 points) Wronskians
Use the Wronskian to prove that the functions
f(x)ex
g(x)
=
X
2
e
h(x)
= e
X
3
are linearly independent on the real line R.
Li
1
I
1
H ci
1+3
iLZ)e
‘I
e
7
C?
/ 11
e(/7/
vidnded
5. (20 points) Nonhomogeneous Linear Equations
Find the solution to the linear ODE:
—
y’
—
= 3x + 4,
with initial conditions
7
y(O)rr_
A
3
y(0)=r_
-
-
(A±) y+
-7A ) AcI
-_/
\/f_
-_7 4-?J1j
V—
X
5
8
_-
—
IPI
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ON
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Si
4
.
I)
(N
\1
+
(N
CD
CD
C
I-.
(ii
(D
C
C
C
C
C
rD