Math 2280 Practice Exam 3
-
University of Utah
Spring 2013
Name:
50
/
+10 fl 5
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) An Endpoint Problem
The eigenvalues for this problem are all nonnegative. First, deter
mine whether ).. = 0 is an eigenvalue; then find the positive eigen
values and associated eigenfunctions.
y”
y’(O)
+ )sy
=
0
=
0;
y’(7r)
=
0.
C)
yQ
AK
(y)D
‘()
/c
y) A
y()
eye,iVcI/e
+h
y(x)
\/
((f)
(4 =
Ic
Ct
(o5() fI(v)
—
(I
(v
1
C c’J
\d)
2
K)
x
0
1
0
n
-
U
N
rn
I
z
(I
•
S
I(
-
1
-7
JI
-7
(I
iI
(I
ii
=)
—,
C
CD
(Th
C
C
rD
2. (10 points) Converting to First-Order Systems
Transform the given differential equation into an equivalent system
of first-order differential equations:
x
3
t
—
2t2xJ+ 3tx’ + 5
;<: Y(
YL
I
Y
1
Y’
(1)
L
3
.÷
(
xI
4
mt
lf 3
(11
3
x
=
vl
1
÷5
4
3. (30 points) Systems of First-Order ODEs
Find the general solution to the system of ODEs:
/0 1 i\
1 0 1 )x.
x’=
f
1 oJ
-Al
-À
C)(--i)+ ()
-À
-
-
-A
3
f]A+L
(A (A(At
-‘
It
()
Ic
;) (
(-j)
qye
IdfeP?6Je4
5
-
More room, if necessary, for Problem 3.
50,
fh
(
50
6
J
1/
4. (20 points) Multiple Eigenvalues’
Find the general solution to the system of ODEs:
‘=(
-j\
uec
-\
5-A
H
(
a
yenVle
o
) er
WC//<f
)
f1ey/)/
ii4eideii4
y
0
Vec
( -I
1
o o
‘This is a hint.
7
J
(
—F
tN
+
)
0
CD
0
0
LI
CD
CD
0
0
0
CD
5. (15 points) Matrix Exponentials
Calculate the matrix exponential eA for the matrix:
0231
0056
0003
0000
C
I
‘
3
0
05
7
Q
/Oo
to
ô
o 1
OC(5)
0cj
0Q
0Oc)
\Oao
0
\ /
)/°°
5
°
o
J(oo&y
oo °°/
O
7Ooo3\/
/Ooco
O
I(o
Oo
00
@0/
00/
33
0000
0
Oc)
0öc
9
0
JTh
cç
Qcc
0Q
c”l
Ii
J
1
j—.
H
0
(0
0
0
(0
(0
0
0
0
(D