Page 850 - Advanced_Engineering_Mathematics o'neil

830 Answers to Selected Problems

⎛ 2t 5t 5t ⎞
e 3e 27te
5t
19. (t) = ⎝ 0 3e 5t (3 + 27t)e ⎠
0 −e 5t (2 − 9t)e 5t
2 3e e 0
⎛ 3t t ⎞
t
⎜0 2e 3t 0 −2e ⎟
1 2e 3t 0 −2e t⎠
21. (t) = ⎝
0 0 0 e t
Section 10.3 Solution of X = AX + G

2 3t
[c 1 (1 + 2t) + 2c 2 t + t ]e
1. X(t) = 2 3t t
[−2c 1 t + (1 − 2t)c 2 + t − t ]e + 3e /2
2 3 6t
[c 1 + c 2 (1 + t) + 2t + t − t ]e
3. X(t) = 2 3 6t
[c 1 + c 2 t + 4t − t ]e
c 2 e
⎛ t ⎞
t
3t
−2c 2 e + (c 3 − 9c 4 )e + e t
⎜ ⎟
2c 4 e 3t ⎠
5. X(t) = ⎝
t
3t
(c 1 − 5c 2 t)e + c 3 e + (1 + 3t)e t

(−1 − 14t)e t
7. X(t) = t
(3 − 14t)e
⎛ 2 −2t ⎞
(6 + 12t + t /2)e
2
(2 + 12t + t /2)e −2t
9. X(t) = ⎝ ⎠
2
3
(3 + 38t + 66t + 13t /6)e −2t
2t 6t 3t
3c 1 e + c 2 e − 4e − 10/3
11. X(t) = 2t 6t
−c 1 e + c 2 e + 2/3
t 7t
c 1 e + 5c 2 e + (68/145)cos(3t) − (54/145)sin(3t) + 40/7
13. X(t) = t 7t
−c 1 e + c 2 e + (2/145)cos(3t) + (24/145)sin(3t) − 48/7
2t
2 + 4(1 + t)e
15. X(t) = 2t
−2 + 2(1 + 2t)e
10cos(t) + t sin(t) − 5t cos(t)
5
17. X = 5 2 5
5cos(t) + sin(t) − t cos(t)
2 2
⎛ 3 1 ⎞
1 2t
− e + (2 + 2t)e − − t
t
4 4 2
e + (2 + 2t)e − 1 − t
⎜ 2t t ⎟
⎠
19. X = ⎝
1
3
2t
5 2t
− e + 2te − − t
4 4 2
Section 10.4 Exponential Matrix Solutions
At
At
In Problems 1 through 5, e is given. The solution is X(t) = e C.
1 1
cos(2t) − sin(2t) sin(2t)
At 2 2
1. e =
5
1
− sin(2t) cos(2t) + sin(2t)
2 2
√ √ √
√ √
3 23 4 23
cos( 23t/2) − sin( 23t/2) − sin( 23t/2)
23
At
3. e = e 13t/2 √ √ √ 23 √ √
8 23 sin( 23t/2) cos( 23t/2) + 3 23 sin( 23t/2)
23 23
⎛ 2 1 3 2t 2 1 1 2t 3 1
cos(t) − sin(t) + e sin(t) + cos(t) − e sin(t) − cos(t) + e
1 2t ⎞
5 5 5 5 5 5 5 5 5
4
4
3 2t
1 2t
At ⎜ 3 cos(t) − sin(t) − e 3 sin(t) + cos(t) + e 7 1 1 2t ⎟
5
5 5 5 5 5 5 5 sin(t) + cos(t) − e ⎠
5
5. e = ⎝
3
1 2t
2
3
1
1 2t
3 2t
− cos(t) − sin(t) + e 1 cos(t) − sin(t) − e 4 cos(t) − sin(t) + e
5 5 5 5 5 5 5 5 5
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October 14, 2010 17:50 THM/NEIL Page-830 27410_25_Ans_p801-866