Note that a given value of
m
may correspond to more than one of the graphs
(with different initial conditions).
(The horizontal axes are time, and the vertical axes are position.)
A.
B.
C.
From the description of the problem, we have
m >
0 .
Graph A can indicate either critical damping or overdamping, so
c
2

4
mk
≥
0 .
Since we are given that
c
= 20 and
k
= 3 , this means 20
2

12
m
≥
0 , so 12
m
≤
400 ,
or
m
≤
100
/
3 . Graph B can also indicate either critical damping or overdamping. In
interval notation, the answer for both A and B is therefore: (0
,
100
/
3] .
Graph C is associated with underdamping, so
c
2

4
mk <
0 . Substituting
c
= 20
and
k
= 3 gives
m >
100
/
3 . In interval notation: (100
/
3
,
∞
) .
3
5.
(20 points) Find all solutions to the differential equation
y
(4)
+ 2
y
000

4
y
00

2
y
0
+ 3
y
= 0
.
The characteristic polynomial is
r
4
+ 2
r
3

4
r
2

2
r
+ 3 . We see that it has 1 as
a root, so it factors as
r
4
+ 2
r
3

4
r
2

2
r
+ 3 = (
r

1)(
r
3
+ 3
r
2

r

3)
= (
r

1)(
r
+ 3)(
r
2

1)
= (
r

1)
2
(
r
+ 1)(
r
+ 3)
.
Therefore a general solution is
y
(
t
) =
c
1
e
t
+
c
2
te
t
+
c
3
e

t
+
c
4
e

3
t
(and the set of all solutions is given by substituting all possible constant real values of
c
1
,
c
2
,
c
3
, and
c
4
into the above expression).
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 Spring '08
 Chorin
 Linear Algebra, Linear Equations, Equations, 12m, linearly independent eigenvectors, dim Nul