(CauchyEuler equation)
Problem FEReview.10m
The Wronskian of the functions
y
1
(
t
) =
e
2
t
and
y
2
(
t
) = 2
t
a
e
2
t
, where
a
is
a real number, is
W
[
y
1
, y
2
] (
t
) = 2
e
4
t
. The value of the parameter
a
is
(a)

1
/
2
(b) 1/2
(c) 1
(d) 3/2
Problem FEReview.11m
A specific springmass system is described by the equation
my
00
+
γy
0
+
ky
= 0
, with mass
m
= 8
kg and spring constant
k
= 2
kg/s
2
.
The value
of the damping coefficient
γ
for which the system changes its behavior from underdamped to
overdamped is:
(a) 6 kg/s
(b) 8 kg/s
(c) 10 kg/s
(d) 12 kg/s
Problem FEReview.12m
e

t
and
e
2
t
are fundamental solutions of the homogeneous equation corre
sponding to
y
00

y
0

2
y
=
te
2
t
. Which of the following is the correct form of the particular
solution (
A
and
B
are two constants)?
(a)
Y
(
t
) =
At
2
e
2
t
(b)
Y
(
t
) =
Ate

t
(c)
Y
(
t
) = (
At
2
+
B
)
e
2
t
(d)
Y
(
t
) = (
At
+
B
)
te
2
t
3
Problem FEReview.13m
Which of the following piecewise continuous functions is drawn on the
picture below?
(a)
f
(
t
) =
t
2
for
0
≤
t <
1
1
.
5
for
1
< t <
2
1
.
375
for
t
= 2
3
.
75

1
.
25
t
for
2
< t
≤
3
(b)
f
(
t
) =
t
2
for
0
≤
t
≤
1
1
.
5
for
1
< t <
2
1
.
375
for
t
= 2
3
.
75

1
.
25
t
for
2
< t
≤
3
(c)
f
(
t
) =
t
2
for
0
≤
t
≤
1
1
.
5
for
1
< t <
2
1
.
25
for
t
= 2
3
.
75

1
.
25
t
for
2
< t
≤
3
(d)
f
(
t
) =
t
2
for
0
≤
t
≤
1
1
.
5
for
1
< t <
2
1
.
375
for
t
= 2
3
.
75

t
for
2
< t
≤
3
Problem FEReview.14m
The Laplace transform
Y
(
s
)
of the solution of the initial value problem
y
00

y
0

2
y
= 0
,
y
(0) = 0
,
y
0
(0) = 2
is:
(a)
Y
(
s
) = 2
/
(
s
2

2
s

2)
(b)
Y
(
s
) = 2
/
(
s
2

2
s

4)
(c)
Y
(
s
) = 2
/
(
s

2)
2
(d)
Y
(
s
) = 2
/
[(
s
+ 1) (
s

2)]
Problem FEReview.15m
The inverse Laplace transform of
F
(
s
) =
5
s
2
+ 2
s
+ 6
is:
(a)
√
5 cos(
√
5
t
)
(b)
√
5
e
t
sin(
√
5
t
)
(c)
√
5
e

t
sin(
√
5
t
)
(d)
√
5 sin(
√
5
t
)
4
Free Response Problems
Problem FEReview.16
Consider the following differential equation:
t
2
dy
dt
=
y

ty
(1)
(a)
Write the equation in the standard form and identify point(s) where the conditions of exis
tence and uniqueness theorem (EUT) are not satisfied.
(b)
Find the the constant solutions, if any. Next, obtain the general solution of the equation.
What is the behavior of the general solution at the point(s) where the conditions of EUT are
not satisfied?
(c)
Find the solutions of the initial value problems: (i)
y
(1) = 1
and (ii)
y
(

1) =

1
and infer
their respective ranges of validity.
(d)
Find the maxima/minima of the solutions (i) and (ii) from subproblem (c), if any. Find the
limit of the solution (i) for
t
→ ∞
and of the solution (ii) for
t
→ ∞
and sketch the
graphs of the solutions in both cases (i) and (ii).
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 Spring '14
 y1