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Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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Jf0
24. y = (b - a)-1 ff [e*(f-5) - ea(f—s)] g(s) ds
t 0
25. y = j-1 f ek(f-s) sini(t — s)g(s) ds
^ 0
26. y = f (t - s)ea(f-s)g(s) ds 29. y = c1t + c2f2 + 4t2ln t
30. y = c11 1 + c2t 5 + 1121 31. y = c1(1 + t) + c2ef + 1 (t — 1)e2f
32. y = c1ef + c2^t — 1 (2t — 1)e f
692
Answers to Problems
See SSM for detailed solutions to 2, 6
8, 9, 12, 17
19, 20, 23, 24
27, 29ac, 30a
1,5, 7ac, 10, 11a
Section 3.8, page 197
1. u = 5 cos(2t - S), S = arctan(4/3) = 0.9273
2. u = 2cos(t — 2๏/3)
3. u = 2v/5cos(3t - S), S = -arctan(1/2) = -0.4636
4. u = cos (๏ t — S), S = ๏ + arctan(3/2) = 4.1244
5. u = 4cos8t ft, t in sec; ๛ = 8 rad/sec, T = n/4sec, R = 1/4 ft
6. u = |sin14t cm, t in sec; t = ๏/14 sec
7. u = (1/4V2) sin(8\/2 t) — 12 cos(8\/2 t) ft, t in sec; ๛ = 8\/2 rad/sec,
T = ๏/4\[ฟ sec, R =/าา/288 = 0.1954 ft, S = ๏ - arctan(3/V2) = 2.0113
8. Q = 10-6cos2000t coulombs, t in sec
9. u = e-10t[2cos(4\/6 t) + (5//6) sin(4\/6 t)] cm, t in sec;
ห = 4\/แ rad/sec, Td = n/2\/6sec, Td/ T = 7/2\[6 = 1.4289, r = 0.4045 sec
10. u = (1/8V3l)e-2t sinpV^T t) ft, t in sec; t = n^^^sec, r = 1.5927 sec
11. u = 0.057198e-015t cos(3.87008 t - 0.50709) m, t in sec; ๋ = 3.87008 rad/sec, ๋/๛0 = 3.87008/VT5 = 0.99925
12. Q = 10-6(2e-500t - e-1000t) coulombs; t in sec
13. ๓ = /20/9 = 1.4907
16. r = s/A2 + B2, rcos0 = B, rsin9 = —A; R = r; S = 0 + (4๋ + 1)๏/2,
n = 0, 1, 2,...
17. ๓ = 8 lb-sec/ft 18. R = 103 ohms
20. v0 < — y u0/2m 22. 2๏/\[3า
23. y = 5 lb-sec/ft 24. * = 6, v = ฑ2^/5
25. (a) r = 41.715 (d) y0 = 1.73, min r = 4.87
(e) r = (2/y) ln(400/V4 - y2)
26. (a) u(t) = e-Yt/2m |^u^4*m — y2 cos ห + (2mv0 + y u0) sin ไ-tj ^4km — y2
(b) R2 = 4m(ku0 + yu0v0 + mv02)/(4*m - y2)
27. plii" + P0gu = ฐ T = 2^/plJp0g
28. (a) u = \/2 sin V2 t (c) clockwise
29. (a) u = (16/V 127)บ-/ sin(V 127 f/8) (c) clockwise
30. (b) u = a cos(/k/m f) + t/m/* sin(/k/m t)
32. (b) u = sin t, A = 1, T = 2n (c) A = 0.98, T = 6.07
(d) e = 0.2, A = 0.96, T = 5.90; e = 0.3, A = 0.94, T = 5.74
(f) e = -0.1, A = 1.03, T = 6.55; e = -0.2, A = 1.06, T = 6.90; e = -0.3, A = 1.11, T = 7.41
Section 3.9, page 205
1. —2sin8tsint 2. 2sin(t/2) cos(13t/2)
3. 2cos(3n t/2) cos(n t/2) 4. 2sin(7t/2) cos(t/2)
5. u" + 256u = 16cos3t, u(0) = 1, u'(0) = 0, u in ft, t in sec
6. u" + 10u' + 98u = 2 sin(t/2), u(0) = 0, u'(0) = 0.03, u inm, t in sec
7. (a) u = 1482 cos 16? + 237 cos 3t (c) ๛ = 16 rad/sec
8. (a) u = า5็า28า[160e-5t cos(v/73't) + e-5t si^V^ t) - 160 cos(t/2) + 3128 sin(t/2)]
(b) The first two terms are the transient. (d) ๛ = 4\/b rad/sec
9. u = 45 (cos7t — cos8t) = า2-sin(t/2) sin(15t/2) ft, t in sec
10. u =(cos8t + sin8f — 8t cos8t)/4ft, t in sec; 1/8, ๏/8, ๏/4, 3n/8sec
11. (a) 981 (30cos2t + sin2t) ft, t in sec (b) m = 4 slugs
12. u = (\/2/6) cos(3t — 3๏/4) m, t in sec
Answers to Problems
693
See SSM for detailed solutions to 15 and 16
22 and 24
CHAPTER 4
2, 8, 13, 17, 19c
21, 27
2, 8, 12
15, 23
15.
F0(t — sin t),
F0[(2n — t) — 3 sin t], —4 F0 sin t,
-1000f
0 < t < n n < t < 2n 2n < t < ๆ
+ 3) coulombs, t in sec, @(0.001) — 1.5468 x 10—
16. Q(t) — 10—6(e—4000f — 4e @(0.01) = 2.9998 x 10—6; Q(t) ^ 3 x 10—6 as t ^ๆ
17. (a) u — [32(2 — a2) cos at + 8a sin at]/(64 — 63a2 + 16a4)
(b) A — 8ฤ/64 — 63a2 + 16a4 (d) a = 3v/14/8 = 1.4031,
A — 64/VT27 — 5.6791
18.
19.
(a) u — 3(cos t — cosat)/(a — 1)
(a) u — [(a2 + 2) cos t — 3 cosaf]/(a2 — 1) + sin t
Section 4.1, page 212
1. —ๆ < t < ๆ 2. t > 0 or t < 0
3. t > 1, or0 < t < 1, or t < 0 4. t > 0
5. ..., —3n/2 < x < —๋/2, —n/2 < x < 1, 1 < x < n/2,
6. —ๆ < x < — 2, — 2 < x < 2, 2 < x < ๆ
7. Linearly independent
8. Linearly dependent; f1(t) + 3 f2(t) — 2 f3(t) — 0
ๆ/2 < x < 3n/2,..
9
10
11
14
17
19
21.
23.
Linearly dependent; 2 f1(t) + 13 f2(t) — 3 f3(t) — 7 f4(f) = 0 Linearly independent
1 12. 1
t = าณ (5) — 1 cos2f
15. 6x
13. —6e—
16. 6/x
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