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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
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e2t + c2e3t + J [e3(t-s) - e2(t-s)] g(s) ds cos2t + c2sin2t + [sin2(t - s)] g(s) ds
13. Y(t) = 1 + t2 ln t 14. Y(t) = —2t2
15. Y(t) = 1 (t - 1)e2t 16. Y(t) = -2(2t - 1)e-t
11. Y(x) = i x2 (ln x)3 18. Y(x) = — x1/2 cos x
7 xet - tex 2 *
19. Y(x) = J (1 - g(t) dt 20. Y(x) = x-1/2J t-3/2sin(x - t)g(t) dt
23. (b) y = y0cos t + yd sin t + / sin(t - s)g(s) ds
Jt0
24. y = (b - a)-1 f1 [eb(t-s) - ea(t-s)] g(s) ds
t 0
25. y = gTl f ek(t-s) sing(t — s)g(s) ds
t 0
26. y = f (t - s)ea(t-s)g(s) ds 29. y = c1t + c2t2 + 4t2ln t
30. y = c11 1 + c2t 5 + 1121 31. y = c1(1 + t) + c2e^ + 1 (t — 1)e2t
32. y = C1 e^ + C2^t — 1 (2t — 1)e ^
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 — 2n/3)
3. u = 2v/5cos(3t — S), S = —arctan(1/2) = —0.4636
4. u = VTl' cos (n t — S), S = n + arctan(3/2) = 4.1244
5. u = 4cos8t ft, t in sec; a = 8 rad/sec, T = n/4sec, R = 1/4 ft
6. u = |sin14t cm, t in sec; t = n/14sec
7. u = (1/4V2) sin(^V2 t) — 12 cos(8\/2 t) ft, t in sec; a = 8\/2 rad/sec,
T = n/4\fl sec, R = VTT/288 = 0.1954 ft, S = n — arctan(3/V2) = 2.0113
8. Q = 10—6cos2000t coulombs, t in sec
9. u = e—10t[2cos(4\/6 t) + (5/V<5) sin(4\/6 t)] cm, t in sec;
H = 4\/6rad/sec, Td = n/2\/6sec, Td/ T = 7/2\[6 = 1.4289, r = 0.4045 sec
10. u = (1/8V3l)e—2t sin(^\/3l t) ft, t in sec; t = n/2\/3lsec, r = 1.5927 sec
11. u = 0.057198e—015t cos(3.87008 t — 0.50709) m, t in sec; ^ = 3.87008 rad/sec,
M/«0 = 3.87008/VT5 = 0.99925
12. Q = 10—6(2e—500t — e—1000t) coulombs; t in sec
13. y = V20/9 = 1.4907
16. r = s/A2 + B2, rcos9 = B, rsin9 = — A; R = r; S = 0 + (4n + 1)n/2,
n = 0, 1, 2,...
17. y = 8 lb-sec/ft 18. R = 103 ohms
20. v0 < —y u0/2m 22. 2n/\f3\
23. y = 5 lb-sec/ft 24. * = 6, v = ±2^/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 nt + (2mv0 + y u0) sin^tj ^4km — y2
(b) R2 = 4m(ku0 + yu0v0 + mv02)/(4*m — y2)
27. plii" + P0gu = 0, T = 2n/pljp0g
28. (a) u = \/2 sin \/2 t (c) clockwise
29. (a) u = (16/V 127)e—sin(V 127 t/8) (c) clockwise
30. (b) u = a cos(^k/m t) + m/k 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 16t + 247 cos 3t (c) a = 16 rad/sec
8. (a) u = i53128T[160e—5t cos(V73 t) + e—5t sin(V73 t) — 160 cos(t/2) + 3128 sin(t/2)]
(b) The first two terms are the transient. (d) a = 4\f3 rad/sec
9. u = 45 (cos7t — cos8t) = H8 sin(t/2) sin(15t/2) ft, t in sec
10. u =(cos8t + sin8f — 8t cos8t)/4ft, t in sec; 1/8, n/8, n/4, 3n/8sec
11. (a) 981 (30cos2t + sin2t) ft, t in sec (b) m = 4 slugs
12. u = (^2/6) cos(3t — 3n/4) m, t in sec
Answers to Problems
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(f - sin f),
F0[(2n - f) - 3 sin f], -4 F0 sin f,
-1000f
0 < f < n n < f < 2n 2n < f < ?
+ 3) coulombs, f in sec, Q(0.001) = 1.5468 x 10-
16. Q(f) — 10-6(e-4000t - 4e Q(0.01) = 2.9998 x 10-6; Q(f) ^ 3 x 10-6 as f
17. (a) u — [32(2 - a2) cos at + 8« sin a>f]/(64 - 63a2 + 16«4)
(b) A — 8/^64 - 63«2 + 16«4 (d) a — 3VT4/8 = 1.4031,
A — 64/VT27 = 5.6791
18. (a) u — 3(cos f - cos af)/(a2 - 1)
19. (a) u — [(a2 + 2) cos f - 3 cos af]/(«2 - 1) + sin f
Section 4.1, page 212
1. -to < f < <x 2. f > 0 or f < 0
3. f > 1, or0 < f < 1, or f < 0 4. f > 0
5. ..., -3n/2 < x < -n/2, -n/2 < x < 1, 1 < x < n/2, n/2 < x < 3n/2,..
6. -to < x < -2, -2 < x < 2, 2 < x < ?
7. Linearly independent
8. Linearly dependent; f1(t) + 3 f2(f) - 2 f3(f) = 0
9
10
11
14
17
19
Linearly dependent; 2 f1(f) + 13 f2(f) - 3 f3(f) - 7 f4(f) = 0 Linearly independent
1 12. 1 13. —6e-
f = 1?(5) - 1 cos2f
15. 6x
16. 6/x
(a) a0[n(n - 1)(n - 2) ••• 1] + a1[n(n - 1) ••• 2]f + •
• + V
(b) (B0Fn + 31rn
• + )er'
(c) ef, e f, b21 , e 2f; yes, W(ef, e f, b21 , e 2f) = 0, -to < f <
00
21. W = ce-2t 23. W = c/f2 27. y = c1et + c2t + c3fef
Section 4.2, page 219
1 V2 e/[(n/4)+2mn ]
3 3ei(n+2mn)
5 2ei[(11n/6)+2mn ]
7. 1, 2 (—1 + i^3), 2 (—1 — i^3)
9. 1 , i2, - 1 , - i 2
11. y = c1et + c2fef + c3e-t
13. y = c1et + c2e2t + c3e-t
15
16
17
18
19
20
22. W = c 24. W = c/f
28. y = c112 + c2f3 + c3(f + 1)
2 2ei [(2n/3)+2mn]
4 [(3n/2)+2mn]
6 4?F [(5n/4)+2mn]
8. 21/4e-ni/8, 21/4e7ni/8
10. (V3 + i)/V2, -(V3 + i)/V2
12. y = c1et + c2fet + c3f2et 14. y = c1 + c2f + c3e2t + c4fe2t
21
f + c2 sin f + ^v^t/2 (c3 cos 2 f + c4 sin 1 f) + e v^3t/2 (c5 cos 1 f + c6 sin 1 f) y = c1et + c2fet + c3f2et + c4e-t + c5fe-t + c6t2e-
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