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MATH  207  Introduction to Differential Equation...
MATH  207  Introduction to Differential Equations  exam Spring 2018 MATH 307 Midterm 2 80 pts total Name: Instruction: • Nothing but writing utensils and a double side 4in × 6in notecard are allowed. • Unless otherwise speciﬁed, you must show work to receive full credit. 1 2 1 (24pts). Find the solution to the initial value problem y 00  y = xe x , y (0) = 1 , y 0 (0) =  1 / 4 The next page is blank in case you need more space for wr
MATH  207  Introduction to Differ...
MATH  207  Introduction to Differential Equations  exam Spring 2018 MATH 307 Midterm 2 80 pts total Name: Instruction: • Nothing but writing utensils and a double side 4in × 6in notecard are allowed. • Unless otherwise speciﬁed, you must show work to receive full credit. 1 2 1 (24pts). Find the solution to the initial value problem y 00  y = xe x , y (0) = 1 , y 0 (0) =  1 / 4 The next page is blank in case you need more space for wr
Page 1
Spring 2018 MATH 307 Midterm 2
80 pts total
Name:
Instruction:
•
Nothing but writing utensils and a double side 4in
×
6in notecard are allowed.
•
Unless otherwise speciﬁed, you must show work to receive full credit.
1
Page 2
2
1 (24pts). Find the solution to the initial value problem
y
00

y
=
xe
x
,
y
(0) = 1
,
y
0
(0) =

1
/
4
The next page is blank in case you need more space for writing.
Page 3
3
(Extra space just in case you need it.)
Page 4
4
2 (12pts). Consider the linear homogeneous equation
t
2
y
00

4
ty
0
+6
y
=0
,
t>
0
.
[a] Find all values of
p
such that
y
(
t
)=
t
p
is a solution to the above equation.
[b] Find the general solution to the diﬀerential equation.
Page 5
5
3. Hanging a mass of
m
=1
kg
stretches the spring by 5
/
(2
π
2
) meters. Use
g
= 10
m/s
2
as the acceleration due to gravity.
[a] (4pts) Find the spring constant
k
.
[b] (4pts) Denote by
y
(
t
) the displacement the mass at any time
t
, with
y
(
t
)
>
0 when
the spring is stretched from the equilibrium position and
y
(
t
)
<
0 when the spring is
compressed. Write down the diﬀerential equation that governs the motion of this un
damped massspring system.
Note: you need to put in actual numbers, not just symbols,
for the mass, spring constant, and etc., to get full credit. No need to show work.
[c] (16pts) Suppose the initial displacement is

1
4
√
2
and the initial velocity is
π
2
√
2
. Find
y
(
t
) for all
t
and express the answer in the form of
y
(
t
)=
A
cos(
ωt

φ
)
.
(I left more space for you to write on the next page in case you need that.)
Page 6
6
3. (Continued)
[d] (6pts) Graph the solution (be sure your graph illustrates the period, amplitude, and
phase shift accurately).
You don’t have to explain.
Page 7
7
4 (14pts). You have a spring in a damping media, but you don’t know the spring con
stant nor the damping constant. To ﬁnd that out, you decide to attach a mass of 1
kg
to the spring and plot the motion of this unforced damped springmass system. The
graph below is a plot of the displacement of the mass at any time
t
. Write down the
diﬀerential equation governing its motion.
Explain your reasoning to get full credit.
Note: you should write down actual (esti
mated) numbers based on what you gather from the graph, not just a symbolic equation.
The next page is blank in case you need more space to work.
Page 8
8
(Extra space in case you need it.)
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