1. Let
y
H
t
L
be defined as the solution of the equation
t y
'+
y
=cos
t
for all
t
>0
satisfying
lim
t
Ø0
y
H
t
L=1
. Then
(a)
y
HpL=1
ÅÅÅÅ
p
(b)
y
HpL=1
(c)
y
Ip
ÅÅÅÅ
2M=2
ÅÅÅÅ
p
(d) None of the above
2. Which of the following statements are true?
(I) The differential equation
dy
ÅÅÅÅÅÅÅ
dt =yety
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1+2
y
2
is separable
(II) The differential equation
y
'' =ty'+2
y
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
y
is nonlinear
(III) All solutions of
y
'' +2
y
=0
are bounded
(IV) The initial value problem,
y
'+sin
t y
= -cos
t
,
y
H0L=1,
y
' H0L=3
has a unique solution
(a) Only (I) and (III)
(b) Only (II) and (III)
(c) Only (II) and (IV)
(d) Only (I) and (IV)
3. An oven is set at a temperature of 324°F. A thermometer initially reading 36°F is placed in the
oven. One minute later the thermometer reads 60°F. How long after it is placed in the oven will the
thermometer read 82°F? [Recall that the rate of change of the temperature of an object is proportional
to the difference between its temperature and the ambient temperature.]
(a) 1.5 minutes
(b) 2 minutes
(c) 3 minutes
(d) None of the above
4. Which of the following functions is NOT contained in the general solution of
y
'''' +2
y
''' +
y
'' -2
y
'-2
y
=0
?
(a)
H1-sin
x
+cos
x
L
e
-
x
(b)
ex
I1-
e
-2
x
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
12 M
(c)
ex
cos
x
-
e
-
x
(d) None of the above
5. For what values of the constants
p
and
q
are all solutions of the equation
y
'' H
t
L+
p y
' H
t
L+
q y
H
t
L=0
periodic functions of
t
?
(a)
p
2=4
q
(b)
p
2<4
q
(c)
p
=0,
q
>0
(d)
p
=0,
q
<0
6. For the equation,
y
'' +2
y
'+2
y
=sin
x
, the method of undetermined coefficients tells us that a particu-
lar solution has the form:
(a)
A
sin
x
(b)
A x
sin
x
(c)
A x
cos
x
+
B x
sin
x
(d)
A
sin
x
+
B
cos
x
7. Using the Euler method to numerically solve
y
' H
t
L-cosHp
t y
L=0,
y
H0L=1
with a step size of .5, the
approximate value for
y
H1L
is
(a)
3
ÅÅÅÅ
2
(b)
1-è!!!!
2
ÅÅÅÅÅÅÅÅÅÅ
4
(c)
3
ÅÅÅÅ
2-è!!!!
2
ÅÅÅÅÅÅÅÅÅÅ
4
(d)
1+è!!!!
2
ÅÅÅÅÅÅÅÅÅÅ
4
8. For what values of the constant
a
does the solution to
y
'' -
y
'-2
y
=18
t e
-
t
,
y
H0L=1,
y
' H0L=
a
remain bounded as
t
Ø ¶
?
(a)
a
= -3
(b)
a
< -1
(c)
a
=0
(d) None of the above
9. Let the function
y
H
t
L
be the solution of the initial value problem
y
'' +4
y
'+4
y
=
e
-2
t
,
y
H0L=0,
y
' H0L=0
. Then:
(a)
y
J1
ÅÅÅÅ
2N=
e
-1
ÅÅÅÅÅÅÅÅ
4
(b)
y
H1L=2
e
-2
(c)
y
H1L=
e
-1
ÅÅÅÅÅÅÅÅ
2
(d)
y
H2L=2
e
-4
10. Write the equation of a harmonic oscillator with spring constant
k
=3
, damping constant
b
=4
and
mass m=1 as a first order system of the form
X
'=
A X
. Then
A
and the general solution of the system are
given by:
(a)
A
=i
k
j
j
j0 1
-3-4y
{
z
z
z
,
X
=
C
1
e
-3
t
i
k
j
j
j1
3y
{
z
z
z+
C
2
e
-
t
i
k
j
j
j-1
1y
{
z
z
z
(b)
A
=i
k
j
j
j0 1
-3-4y
{
z
z
z
,
X
=
C
1
e
-3
t
i
k
j
j
j-1
3y
{
z
z
z+
C
2
e
-
t
i
k
j
j
j1
-1y
{
z
z
z
(c)
A
=i
k
j
j
j0 1
4 3 y
{
z
z
z
,
X
=
C
1
e
4
t
i
k
j
j
j1
4y
{
z
z
z+
C
2
e
-
t
i
k
j
j
j-1
1y
{
z
z
z
(c)
A
=i
k
j
j
j0 1
4 3 y
{
z
z
z
,
X
=
C
1
e
4
t
i
k
j
j
j-1
4y
{
z
z
z+
C
2
e
-
t
i
k
j
j
j1
-1y
{
z
z
z
11. Suppose that
y
H
t
L
satisfies the boundary value problem,
y
'
y
'' =
t
,
y
' H0L=0,
y
H1L=0
. Which of the
following statements is true?
(a)
y
H
t
L=0
(b) y(t) is unique
(c)
y
H
t
L
does not exist
(d)
y
H
t
L
is monotone on [0,1]
12. Suppose that the general solution of
y
'' H
t
L+
f
H
t
L
y
' H
t
L+
g
H
t
L
y
H
t
L=0
is given by
y
H
t
L=
C
1
t
2+
C
2
t
-2
.
A particular solution of
y
'' +
f y
'+
g y
=
t
2
is
(a)
yp
=
t
2
ÅÅÅÅÅÅÅ
12
(b)
yp
=12
ÅÅÅÅÅÅÅ
t
2
(c)
yp
=
t
4
ÅÅÅÅÅÅÅ
6
(d) None of the above
2214-F04-cte.nb
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