Page 828 - Advanced_Engineering_Mathematics o'neil

808 Answers to Selected Problems

1 s 1
3. (1 − e −7s ) + cos(7)e −7s − sin(7)e −7s
2
2
s s + 1 s + 1
1 11 4
5. − e −3s − e −3s
s 2 s s 2
1 2 1
7. − +
s + 1 (s + 1) 3 (s + 1) + 1
2
s 2 s 1
9. + − − e −2πs
2
2
2
s + 1 s s + 1 s + 1
2
(s + 1) − 9
11.
2
((s + 1) + 9) 2
1 2 1 1
13. − − + e −16s
s 2 s s 2 s
24 4 1
15. + +
(s + 5) 5 (s + 5) 3 (s + 5) 2
1
2t
17. F(s) = , so f (t) = e sin(t)
(s − 2) + 1
2
1
19. sin(t − 2)H(t − 2)
3
1 1 √
21. F(s) = , so f (t) = √ e −3t sinh 2t
2
(s + 3) − 2 2
(s + 3) − 1
23. F(s) = so
(s + 3) − 8
2
√ 1 √
f (t) = e −3t cosh 2 2t − √ e −3t sinh 2 2t
2 2
1
25. (1 − cos(4(t − 2)))H(t − 2)
16
3
27. y = cos(2t) + (1 − cos(2(t − 4)))H(t − 4)
4

1 1 1 √
29. y = − + e −2(t−6) + e −(t−6) cos 3(t − 6) H(t − 6)
4 12 6
1 2 3 1
t
31. y =− + e − cos(2t) − sin(2t)
4 5 20 5

1 2 3 1
− − + e t−5 − cos(2(t − 5)) − sin(2(t − 5)) H(t − 5)
4 5 20 5
33. E out = 5e −4t + 10[1 − e −4(t−5) ]H(t − 5)
k k
35. i(t) = (1 − e −Rt/L ) − (1 − e −R(t−5)/L )H(t − 5)
R R
Section 3.4 Convolution
1
1. [sinh(2t) − sin(2t)]
16
cos(at) − cos(bt)
2
2
2
3. if b
= a ;t sin(at)/2a if b = a 2
(b − a)(b + a)
1 1
5. [1 − cos(at)]− t sin(at)
a 4 2a 3
1 1

7. − e −2(t−4) H(t − 4)
2 2
2t
3t
9. y(t) = e ∗ f (t) − e ∗ f (t)
1 1
6t
6t
2t
11. y(t) = e ∗ f (t) − e ∗ f (t) + 2e − 5e 2t
4 4
1 1
13. y(t) = sin(3t) ∗ f (t) − cos(3t) + sin(3t)
3 3
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October 14, 2010 17:50 THM/NEIL Page-808 27410_25_Ans_p801-866