Page 829 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 809

4 1 1 1
t
2t
t
15. y(t) = e − e − e −2t − e ∗ f (t)
3 4 12 3
1 1
2t
+ e ∗ f (t) + e −2t ∗ f (t)
4 12
1 3
17. f (t) = e −2t − 19. f (t) = cosh(t)
2 2
2 √ √
21. f (t) = 3 + 15e t/2 sin 15t/2
5
23. r(t) = Akt
1 1

2
25. r(t) = (Ak + B)t + kB + C t + kCt 3
2 3
Section 3.5 Impulses and the Delta Function
1. y(t) = 3[e −2(t−2) − e −3(t−2) ]H(t − 2) − 4[e −2(t−5) − e −3(t−5) ]H(t − 5)
t
−t
3. y = 6(e −2t − e + te )
5. ϕ(t) = (B + 9)e −2t − (B + 6)e −3t

m k

7. y(t) = v 0 sin t
k m
Section 3.6 Solution of Systems
t/2
1. x(t) =−t − 2 + 2e , y(t) =−t − 1 + e t/2
1 4 2
3. x(t) = t + (1 − e 3t/4 ), y(t) = (−1 + e 3t/4 )
3 9 3
1 3 3
2
5. x(t) = (t + t ) + (1 − e 2t/3 ), y(t) = t + (1 − e 2t/3 )
2 4 2
−t
−t
2
7. x(t) = t − 1 + e cos(t), y(t) = t − t + e sin(t)
−t
9. x(t) = 1 − e − 2te , y(t) = 1 − e −t
−t
1 1 1 1 1
2
−t
t
−t
t
11. y 1 (t) =−1 − t + (e + e ), y 2 (t) =− t − t , y 3 (t) =− t − (e − e )
2 2 4 3 6
13. The loop currents satisfy

5i − 5i 1 − 5i = 1 − H(t − 4)sin(2(t − 4)),

1 2
−5i + 5i + 5i 2 = 0.

1
2
The currents are
1 1 −t/2
i 1 (t) = 1 − e
5 2
2 9
− e −(t−4)/2 − cos(2(t − 4)) + sin(2(t − 4)) H(t − 4)
85 2
1 2
i 2 (t) = e −t/2 + e −(t−4)/2
10 85
−cos(2(t − 4)) − 4sin(2(t − 4))) H(t − 4).
15. The system is

x + 8x 1 − 2x 2 = 1 − H(t − 2),−2x 1 + x + 5x 2 = 0.
2
1
The solution is
5 1 4
x 1 (t) = − cos(2t) − cos(3t)
36 20 45

5 1 4
− − cos(2(t − 2)) − cos(3(t − 2)) H(t − 2),
36 20 45
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October 14, 2010 17:50 THM/NEIL Page-809 27410_25_Ans_p801-866