Chapter 5
5.
In this problem,
C
=0
.
01 F,
L
=4H
,and
R
= 10 Ω. Hence, the equation governing the
RLC
circuit is
4
d
2
I
dt
2
+10
dI
dt
+
1
0
.
01
I
=
d
dt
(
E
0
cos
γt
)=
−
E
0
γ
4
sin
γt.
The frequency response curve
M
(
γ
) for an
RLC
curcuit is determined by
M
(
γ
)=
1
p
[(1
/C
)
−
Lγ
2
]
2
+
R
2
γ
2
,
which comes from the comparison Table 5.3 on page 290 of the text and equation (13) in
Section 4.9. Therefore
M
(
γ
)=
1
p
[(1
/
0
.
01)
−
4
γ
2
]
2
+ (10)
2
γ
2
=
1
p
(100
−
4
γ
2
)
2
+ 100
γ
2
.
The graph of this function is shown in Figure B.43 in the answers of the text.
M
(
γ
)hasits
maximal value at the point
γ
0
=
√
x
0
,whe
re
x
0
is the point where the quadratic function
(100
−
4
x
)
2
+ 100
x
attains its minimum (the ±rst coordinate of the vertex). We ±nd that
γ
0
=
r
175
8
and
M
(
γ
0
)=
2
25
√
15
≈
0
.
02
.
7.
This spring system satis±es the di²erential equation
7
d
2
x
dt
2
+2
dx
dt
+3
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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 Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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