Page 827 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 807

6

4

2

0
0 2 4 6 8 10 12 14
t
–2

FIGURE A.7 Graphs in Problem 17, Section 2.4

CHAPTER THREE THE LAPLACE TRANSFORM
Section 3.1 Definition and Notation
2
s − 4
1. 3
2
(s + 4) 2
14 7
3. −
s 2 s + 49
2
1 3
5. −10 +
2
(s + 4) 3 s + 9
3 −3t
7. cos(8t) 9. e −42t − t e /6
5 1
15. L[ f ](s) =
s 1 + e −3s
5e −5s (1 − e −5s )
17. L[ f ](s) =
s(1 − e −25s )
Eω 1
19.
2
2
s + ω 1 − e −πs/ω
h
21. L[ f ](s) =
s(1 + e −as )
Section 3.2 Solution of Initial Value Problems
1 13
1. y = − e −4t
4 4
4 4 1
3. y =− e −4t + cos(t) + sin(t)
17 17 17
1 1 17
5. y =− + t + e 2t
4 2 4
22 13 3 4
2t
2t
7. y = e − te + cos(t) − sin(t)
25 5 25 25
1 1 33 15
9. y = + t − cos(4t) + sin(4t)
16 16 16 64
Section 3.3 Shifting and the Heaviside Function
6 3 2
1. − +
(s + 2) 4 (s + 2) 2 s + 2
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October 14, 2010 17:50 THM/NEIL Page-807 27410_25_Ans_p801-866