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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.
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9. (a) dv/dt = 9.8, v(0) = 0 (b) T = V300/4.9 = 7.82 sec
(c) v = 76.68 m/sec
10. (a) r = 0.02828 day—1 (b) Q(t) = 100e—0 02828t (c) T = 24.5 days
12. 1620ln(4/3)/ ln2 = 672.4 years
13. (a) Q(t) = CV(1 — e—t/RC) (b) Q(t) ^ CV = QL
(c) Q(t) = CVexp[— (t — t1)/RC]
14. (a) Q = 3(1 — 10—4Q), Q(0) = 0
(b) Q(t) = 104(1 — e—3t/10 ), t inhrs; after 1 year Q = 9277.77 g
(c) Q = — 3 Q/104, Q(0) = 9277.77
(d) Q(t) = 9277.77e—3t/10 , t inhrs; after 1 year Q = 670.07 g
(e) T = 2.60 years
15. (a) q' = —q/300, q(0) = 5000 g (b) q(t) = 5000e—t/300 (c) no
(d) T = 300 ln(25/6) = 7.136 hr
(e) r = 250ln(25/6) = 256.78 gal/min
Section 1.3, page 22
2, 6, 8, 14, 16, 19 1. Second order, linear 2. Second order, nonlinear
3. Fourth order, linear 4. First order, nonlinear
5. Second order, nonlinear 6. Third order, linear
15. r =-2 16. r =±1
17. r = 2, ---3 18. r = 0, 1, 2
19. r =-1, -2 20. r = 1, 4
22, 26 21. Second order, linear 22. Second order, nonlinear
23. Fourth order, linear 24. Second order, nonlinear
CHAPTER 2 Section 2.1, page 38
1. (c) y = ce—3t + (t/3) — (1 /9) + e—2t; y is asymptotic to t/3 — 1/9 as t ^ æ
1,2,3 2. (c) y = ce2' + t3e2t/3; y —> æ as t —> æ
3. (c) y = ce—t + 1 + t2e—t/2; y — 1 as t — æ
681
See SSM for detailed solutions to 4, 6, 7, 11, 13, 15, 18, 20.
21b, 24bc, 28
30, 32, 35, 36, 37
1,4
6, 10ac, 13, 15, 17a
17c, 19ac
4.
5.
6.
7
8 9
10
11
12
13
15
17
19
21
22.
23.
25.
27.
28. 29.
36.
37.
(c) 7 = (c/t) + (3 cos2t)/4t + (3 sin2t)/2; 7 is asymptotic to (3 sin2t)/2 as t ¦
(c) 7 = ce2t — 3et; 7 ^æ or -æ as t ^ æ
(c) 7 = (c - tcos t + sin t)/12; 7 ^ 0 as t ^ æ 2 -2 -2 (c) 7 = t2e 1 + ce 1 ; 7 ^ 0 as t ^ æ
(c) 7 = (arctan t + c)/(1 + t2)2; 7 ^ 0 as t ^ æ
(c) 7 = ce-1/2 + 3t - 6; 7is asymptotic to 3t - 6 as t ^ æ
(c) 7 =—te-t + ct; 7 ^æ, 0, or -æ as t ^ æ
(c) 7 = ce-t + sin2t - 2 cos2t; 7 is asymptotic to sin2t - 2 cos2t as t ^ æ
(c) 7 = ce-t/2 + 3t2 — 121 + 24; 7 is asymptotic to 3t2 - 12t + 24 as t ^ æ
7 = 3er + 2(t - 1)e2t
7 = (3t4 - 4¥3 + 6t2 + 1)/12t2
7 = (t + 2)e2t
7 =-(1 + t)e-‘/t\ t = 0
(b) 7 = —4 cos t + 5 sin t + (a + 4)et/2;
5 vv/o «. I 5
(c) 7 oscillates for a = a0
(b) 7 = -3et/3 + (a + 3)et/2;
(c) 7 ^ -æ for a = a0
(b) 7 = te-t + (ea — 1)e-t/1;
(c) 7 ^ 0 as t ^ 0 for a = a0
14.
16.
18.
20.
a0 =
= (t2 - 1)
e 2t/2 2
7
7 = (sin t)/t2
7 = t 2[(n2/4) - 1 - tcos t + sin t] 7 = (t - 1 + 2e-‘)/t, t = 0
a3
a0 = 1 /e
24. (b) 7 =-cos t/12 + n2a/4t2; a0 = 4/n2
(c) 7 ^ ³ as t ^ 0 for a = a0 (t, 7) = (1.364312, 0.820082) 26.
(a) 7 = 12 + 65 cos2t + gsin2t - 75 e-t/4;
65 + 65
(b) t = 10.065778 70 = -5/2
70 = -16/3; 7 ^-t^as t-
See Problem 2.
See Problem 4.
70 = -1.642876 7 oscillates about 12 as t
æ for 70 = —16/3
Section 2.2, page 45
1.
2.
3.
4.
5.
6.
7.
8. 9.
11.
13.
15.
17.
19.
3 ó2 - 2x3 = c; 7 = 0
3 T2 — 2ln|1 + x31 = c; x = — 1, 7 = 0
7-1 + cos x = c if 7 = 0; also 7 = 0; everywhere
3 7 + 72 - x3 + x = c; 7 = -3/2
2tan27 — 2x — sin2x = c if cos27 = 0; also 7 = ±(2n + 1)n/4 for any integer n; everywhere
7 = sin[ln |x| + c] if x = 0 and 17I < 1; also 7 =±1
j2 — x2 + 2(e7 — e-x) = c; 7 + e7 = 0
3 7 + 73 - x3 = c; everywhere
(a) 7 = 1/(x2 - x - 6)
(c) - 2 < x < 3
(a) 7 = [2(1 - x)ex - 1]1/2
(c) —1.68 < x < 0.77 approximately
(a) 7 = — [2ln(1 + x2) + 4]1/2
(c) -æ < x < æ
(a) 7 = — + 5\/4x2 - 15
(c) x > 2vT5 _________________
(a) 7 = 5/2 -Óx3 - ex + 13/4
(c) — 1.4445 < x < 4.6297 approximately
(a) 7 = [n — arcsin(3 cos2 x)]/3
(c) |x - n/2| < 0.6155
10.
12
1/2
(a) 7 = —-/2x — 2x2 + 4 (c) - 1 < x < 2 (a) r = 2/(1 - 2 ln0)
(c) 0 <9 < ë/e___________
14. (a) 7 = [3 - 2^1 + x2]"
(c) |x| < 2^5 16. (a) 7 = — (x2 + 1)/2 (c) -æ < x < æ (a) 7 = — + ³^65 - 8ex - 8e (c) |x| < 2.0794 approximately (a) 7 = [|(arcsinx)2]1/3
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