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Math 205, Spring 2011
Homework 12:
due April 18
Chapter 6, Sections 5, 6 and 7.
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Week 12:
6.5 Springs
6.6 Circuits
6.7 Variation of Parameters
The general homogeneous massspring system is
m
d
2
y
dt
2
+
c
dy
dt
+
ky
= 0
,
where
m
is the mass,
k
is the spring constant and
c
is
the damping constant (also called “friction”). In the case
c
= 0
the system is said to be
undamped.
When
c
n
= 0 the system is
damped,
and the motion depends primarily
on the type of damping, determined by the roots.
Problem 5
Solve
y
′′
+ 2
y
′
+ 5
y
= 0
,
with IV
y
(0) = 1
, y
′
(0) = 3
,
and determine whether the system
is
overdamped, critically damped
or
damped and oscillating,
describe the sketch.
(soln.) char. polyn.
r
2
+ 2
r
+ 5 = 0
,
roots are
−
1
±
2
i,
complex, so the system is underdamped. With
y
=
e
−
t
(
c
1
cos 2
t
+
c
2
sin 2
t
) we get 1 =
y
(0) =
c
1
,
y
=
e
−
t
(cos 2
t
+
c
2
sin 2
t
)
,
so
y
′
=
−
e
−
t
(cos 2
t
+
c
2
sin 2
t
) +
e
−
t
(
−
2 sin 2
t
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This note was uploaded on 11/24/2011 for the course MATH 205 taught by Professor Zhang during the Spring '08 term at Lehigh University .
 Spring '08
 zhang
 Math

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