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Exercises 5.5
The frst equation yields
z
=
−
x
+
1
k
(
mD
2
+2
k
)
2
[
x
]=
1
k
2
(
mD
2
k
)
2
−
1
[
x
]
,
(5.30)
and so
−
(
mD
2
k
)
[
x
]+
(
mD
2
k
)
±
1
k
2
(
mD
2
k
)
2
−
1
[
x
]
²
=
(
mD
2
k
)
1
k
2
(
mD
2
k
)
2
−
2
[
x
]=0
.
The characteristic equation For this homogeneous linear equation with constant coeﬃcients is
(
mr
2
k
)
1
k
2
(
mr
2
k
)
2
−
2
=0
,
which splits onto two equations,
mr
2
k
⇒
r
=
±
i
r
2
k
m
(5.31)
and
1
k
2
(
mr
2
k
)
2
−
2=0
⇒
(
mr
2
k
)
2
−
2
k
2
⇒
³
mr
2
k
−
√
2
k
´³
mr
2
k
+
√
2
k
´
⇒
r
=
±
i
s
(2
−
√
2)
k
m
,r
=
±
i
s
(2 +
√
2)
k
m
.
(5.32)
Solutions (5.31) and (5.32) give normal Frequences
ω
1
=
1
2
π
r
2
k
m
,ω
2
=
1
2
π
s
(2
−
√
2)
k
m
3
=
1
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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