Download Math 2280-001 Quiz 11 SOLUTIONS April 17, 2015

Math 2280-001
Quiz 11 SOLUTIONS
April 17, 2015
1a) Use the Laplace transforms to solve the IVP
F0
2
x##C w0 x t =
cos w0 t
m
x 0 = x0
x# 0 = v0
(5 points)
Solution: The differential equation is an equality for the actual solution x t , so the Laplace transforms of
both sides also agree:
F0
2
s
s2 X s K sx0 K v0 C w0 X S =
m s2 C w2
0
2
s2 C w0 =
X s
X s =
F0
m
s
F0
s
x t =
2
2 2
C w0
F0
C s x0 C v0
m s2 C w2
0
C x0
t
sin w0 t
2 w0
m
s
s
s
2
2
C w0
C
w0
v0
w0 s2 C w2
0
C x0 cos w0 t C
v0
w0
.
sin w0 t .
1b) What phenomenon do solutions to 1a exhibit?
(1 points)
Resonance - the first term in the solution above has linearly growing amplitude.
2) Use the table to compute the inverse Laplace transform LK1 F s
t for
5
3
4s
F s =
C
C 2
.
2
s
sK3
s C 2 s C 10
(4 points)
solution:
F s =
5
C
s
3
sK3
2
C
4 sC1
sC1
2
C9
K
4
3
3
2
sC1
so
x t = 5 C 3 t e3 t C 4 eKt cos 3 t K
4 Kt
e sin 3 t .
3
C9