Math 201 Fall 2011 Midterm Exam 3 (B)
November 29, 2011
1. [5] Set up (but do not solve) the diﬀerential equation of motion for a springmass system
where the mass is 2 units, the damping coeﬃcient is 2 and the spring constant is 2 with an
external applied force
F
(
t
) =
0
,
0
≤
t <
1
,
2
,
1
≤
t <
2
,
0
, t >
2
,
using the unit step function;
•
The equation is
mx
00
+
γx
0
+
kx
=
F
(
t
)
, where the mass
m
= 2
, the spring constant
k
= 2
, the damping coeﬃcient
γ
= 2
.
•
F
(
t
) = 2[
U
(
t

1)

U
(
t

2)]
.
•
So, the equation is
2
x
00
+ 2
x
0
+ 2
x
= 2
U
(
t

1)

2
U
(
t

2)
or, simplify it,
x
00
+
x
0
+
x
=
U
(
t

1)

U
(
t

2)
2. [5] Find the general solution of
x
2
y
00
+ 3
xy
0
+ 1
y
= 0
.
•
The auxiliary equation is
m
(
m

1)+3
m
+1 =
m
2
+2
m
+1 = (
m
+1)
2
= 0
, a repeated
root
m
=

1
.
•
Thus the two linearly independent solutions are
y
1
=
x

1
and
y
2
=
y
1
ln
x
=
x

1
ln
x
.
•
The general solution is thus
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