Page 825 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 805

5
5
4
4
3
3

2
2
1 1

0
0
0 1 2 3 4 5
0 2 4 6 8 10 t
t –1
FIGURE A.1 Graphs in Problem 1, Section FIGURE A.2 Graphs in Problem 3, Section 2.4
2.4

2
1.5
1.5
1
1
0.5
0.5
0 0
0 2 4 6 8 0 1 2 3 4
–0.5 t –0.5 t

–1
–1

FIGURE A.3 Graphs in Problem 5, Section 2.4 FIGURE A.4 Graphs in Problem 7, Section 2.4

3. With y(0) = 5and y (0) = 0,

5
−t
y = e [2cos(2t) + sin(2t)]
2

and with y(0) = 0and y (0) = 5,
5
−t
y = e sin(2t).
2
Graphs are in Figure A.2.
√
A
5. y = √ e −2t sinh( 2t); graphs are in Figure A.3, proceeding from the lowest to the highest graph as A increases.
2
7. y = Ate −2t , with the graphs in Figure A.4 moving from the lowest to the highest as A increases.
A −t
9. y = e sin(2t), with the graphs moving from lowest to highest as A increases. Graphs are given in Figure A.5,
2
increasing as A increases.
11. At most once; a condition on y(0) alone is not enough to guarantee that the bob never passes through the origin. A

condition on y (0) is also needed.
√
2
13. Obtain ω = 4km − c /2m, so increasing c decreases the frequency ω.
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October 14, 2010 17:50 THM/NEIL Page-805 27410_25_Ans_p801-866