Solutions to exercises 643
Section 12.8
34.y 1 = 1 ae
2 x1 + 1 (b 1 − 1 x)e
−x35.y 1 = 1 (a 1 + 1 bx 1 + 1 x
2
2 2)e
4 x37.y 1 = 1 (a 1 − 13 x 2 4) 1 cos 12 x 1 + 1 b 1 sin 12 x
- (ii)I
h(t) 1 = 1 ae
−(α+β)t1 + 1 be
−(α−β)twhereα 1 = 1 R 22 L,β
2
1 = 1 α
2
1 − 1 (1 2 LC)
Chapter 13
Section 13.2
= 1 ae
3 x1 + 1 be
− 3 xifa
0
1 = 1 a 1 + 1 b, a
1
1 = 1 3(a 1 − 1 b)
Section 13.3
6.r
1
1 = 1 r
2
1 = 1 − 1 7.r 1 = 1 ±n
8.r
1
1 = 1 r
2
1 = 10 9.r
1
1 = 1 − 2 ,r
2
1 = 1 − 3
- (i) (ii)y 1 = 1 (a 1 + 1 b 1 ln 1 x)x
ryax bx
rr=+
12++
!
⋅
!
⋅⋅
!
ax x x x
1
47 10
2
4
25
7
258
10
ya=+x x x
!
⋅
!
⋅⋅
!
0
36 9
1
1
3
14
6
147
9
yaxa x
xxx
=+ −−
−−−
10
2
468
1
357
!
!
!
a xx x
1
35
3
3
1
3
3
3
5
() () ()
ya
xx
=+
!
!
0
24
1
3
2
3
4
() ()
ya x
mm=
=
∑
00
∞
ya x m ae
mmx==
=
∑
03
0
0
3!
∞
−+
1
39
(cos sin )53 3xx
yae be
x
e
xx x=+ −−+
32 −− 3
1
42
1
3
yae be x x
xx=+ − +
32 −
1
39
(cos sin )53 3
yae be e
xx x=+ +
32 3−−
1
3
yabxe x x x
x=+() (− + + +)
423
1
32
19 12 8
yae be
x
xx=+ −−
32 −
1
42
y
x
p=− −
1
42
11.y 1 = 1 ax
122
1 + 1 bx 12.y 1 = 1 (a 1 + 1 b 1 ln 1 x)x
13.y 1 = 1 ax
− 122
1 cos 1 x 14.y
1
1 = 1 ae
xSection 13.4
- (i)P
1
1
1 = 1 sin 1 θ
(ii)
P
4
3
1 = 11051 sin
3
1 θ 1 cos 1 θ,P
4
4
1 = 11051 sin
4
1 θ
Section 13.5
- (i)H
5
1 = 132 x
5
1 − 1160 x
3
1 + 1120 x
(iii)H
6
1 = 164 x
6
1 − 1480 x
4
1 + 1720 x
2
1 − 1120
Section 13.6
- (i)
24.L
4
1 = 1 x
4
1 − 116 x
3
1 + 172 x
2
1 − 196 x 1 + 124
Section 13.7
- (i)
(ii)
Chapter 14
Section 14.2
2.c 2 D 1 = 1 0: V 1 = 1 A 1 + 1 Bx
c 2 D 1 = 1 λ
2
1 > 1 0: V 1 = 1 Ae
λx1 + 1 Be
−λxc 2 D 1 = 1 −λ
2
1 < 1 0: V 1 = 1 A 1 cos 1 λx 1 + 1 B 1 sin 1 λx
J
x
xx
x
x
−
=−
52
2
23
1
3
π
cos sin
J
x
xx
x
x
52
2
23
1
3
=−
−
π
sin cos
−
6
1
35 2!!
x
J
xxx
2
22
2
1
2
1
13 2
1
24 2
=
−
!!! !!
4
−
−−
nn n
x
()( )
(!)
12
3
2
3
ya
n
x
nn
=− x
−
0
22
2
1
1
1
(!) 2
()
(!)
P
4
222
15
2
=−sin ( cosθθ 71 )
P
4
13
5
2
=−sin ( cosθθθ 73 cos )
Px x x x
6
642
1
16
() (=−+−231 315 105 5)
ya
x
x
b
x
x
=+
cos sin 22