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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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(c) - 1 < 2x < 1
18
20
682
Answers to Problems
See SSM for detailed solutions to 21, 23.
25,27ab 29,30bd, 31,33
35b
1, 2, 3
4, 7abc, 9, 10, 12a, 16a
21. y3 - 3y2 - x - x3 + 2 = 0, |x| < 1
22. y3 - 4y - x3 = -1, Ix3 - 1| < 16/3\/3 or-1.28 < x < 1.60
23. y = -1/(x2/2 + 2x - 1); x =-2
24. y =-3/2 + Ó2x - ex + 13/4; x = ln2
25. y =—3/2 + ^sin2x + 1/4; x = n/4
26. y = tan(x2 + 2x); x = -1
27. (a) y ^ 4 if Óî > 0; y = 0 if yo = 0; y ^-æ if yo < 0
(b) T = 3.29527
28. (a) y ^ 4 as t
(b) T = 2.84367
(c) 3.6622 < y0 < 4.4042
c ad — bc
29. x = — y +-----------r— ln |ay + b| + k; a = 0, ay + b = 0
a a2
30. (e) |y + 2x|3|y - 2x| = c
31. (b) arctan(y/x) - ln |x| = c
32. (b) x2 + y2 - cx3 = 0
33. (b) |y- x| = c|y + 3x|5; also y = -3x
34. (b) |y + x||y + 4x|2 = c
35. (b) 2x/(x + y) + ln |x + y| = c; also y =-x
36. (b) x/(x + y) + ln |x| = c; also y =-x
37. (b) |x|3|x2 - 5y2| = c
38. (b) c|x|3 = |y2 - x21
Section 2.3, page 57
1. t = 100ln100min = 460.5min
2. Q(t) = 120y[1 - exp(— t/60)]; 120y
3. Q = 50e-02(1 - e-0'2) lb ^ 7.42 lb
4. Q(t) = 200 + t - [100(200)2/(200 + t)2] lb, t < 300; c = 121/125 lb/gal;
lim c = 1 lb/gal
5. (a) Q(t) = 62s0T0e-t/50 + 25 - 2m cos t + 50ù sin t
(c) level = 25; amplitude = 2^2501/5002 = 0.24995
6. (a) (ln2)/r years (b) 9.90 years (c) 8.66%
7. (a) k(ºÃ³ - 1)/r (b) k = \$3930 (c) 9.77%
8. (a) A: \$337,733.85; B: \$250,579.41
(b) A: 2000e30r(e10r - 1)/r; B: 2000(e30r - 1)/r
(d) r = 0.0609
9. k = \$3086.64/year; \$1259.92
10. (a) \$89,034.79 (b) \$102,965.21
11. (a) \$99,498.08 (b) \$188,501.92
12. (a) t = 135.36 months
(b) \$152,698.56
13. (a) (k/r) + [S0 - (k/r)]ert (b) rS0 (c) (1/r) ln[k/(k - k0)] years
(d) T = 8.66 years (e) rS0erT/(erT - 1) (f) \$119,716
14. (a) 0.00012097 year-1 (b) Q0 exp(-0.00012097t), t in years
(c) 13,305 years
15. P = 201,977.31 - 1977.31 e(ln2)t, 0 < t < t{ = 6.6745 (weeks)
16. (a) t = 2.9632; no
(b) t = 10 ln2 = 6.9315
(c) t = 6.3805
17. (b) yc = 0.83
18. t = ln!3/ln Ö min = 6.07min
Answers to Problems
683
See SSM for detailed solutions to 19, 20ab, 21 ab, 22,24ab
24cd, 26ab, 27ab, 28
29b, 30, 31bdef
1,4, 8, 11, 13, 17
22abc
19. (a) x = 0.04[1 — exp(— t/12,000)] (b) ò = 36min
0 — k — (P/r)]e-rt/ V; lim c = k + (P/r)
0 t—æ
(b) T = (Vln2)/r; T = (Vln10)/r
(c) Superior, T = 431 years; Michigan, T = 71.4 years; Erie, T = 6.05 years; Ontario, T = 17.6 years
21. (a) 50.408 m (b) 5.248 sec 22. (a) 45.783 m (b) 5.129 sec
23. (a) 48.562 m (b) 5.194 sec
24. (a) 176.7 ft/sec (b) 1074.5 ft (c) 15 ft/sec (d) 256.6 sec
25. (a)
ln 1 +
kv
+
kv
k2 V mg J k m k \ mg ó
v = —(mg/k) + [v0 + (mg/k)] exp(—kt/m) (b) v = v0 — gt; yes
v = 0 for t > 0
vL = 2a2g(p — p')/9ä (b) e = 4n a3g(p — p')/3 E
11.58 m/sec (b) 13.45m (c) k > 0.2394 kg/sec
29. (a) v = R^2g/(R + x) (b) 50.6 hr
(1+1 Ó'
31. (b) x = utcos A, y = —gt2/2 + ut sin A + h
(d) —16L2/(u2 cos2 A) + L tan A + 3 > H
(e) 0.63 rad < A < 0.96 rad
(f) u = 106.89 ft/sec, A = 0.7954 rad
32. (a) v = (u cos A)e—rt, w = —g/r + (u sin A + g/r)e—rt
x = u cos A(1 — e—rt)/r, y = —gt/r + (u sin A + g/r)(1 — e—rt)/r + h
u 145.3 ft/sec, A 0.644 rad
26. (a) (c)
27. (a)
28. (a)
30. v
altitude = 1536 miles
(b)
(d)
33. (d) k = 2.193
Section 2.4, page 72
14
15
2. 0 < t < 4 4. —æ < t < —2 6. 1 < t < n 8. t2 + y2 < 1 t = 0, y = 0
11. y = 0, y = 3
1. 0 < t < 3
3. æ/2 < t < 3n/2
5. -2 < t < 2
7. 2t + 5y > 0or2t + 5y < 0
9. 1 — t2 + y2 > 0 or 1 — t2 + y2 < 0,
10. Everywhere
12. t = nn for n = 0, ±1, ±2,...; 13. y = ±^Ó — 4t2 if y0 = 0;
Ó =—1 |t| < |y0|/2
Ó = [(1/Ó0) — t2]—1 if y0 = 0; y = 0 if y0 = 0; interval is |t| < 1/JT0 if y0 > 0; -æ < t < æ if y0 < 0
Ó = yJ^y+X if Ó0 = 0; y = 0 if Ó0 = 0; interval is
— 1 /2Ó2 < t < æ if y0 = 0; —æ < t < æ if y0 = 0
y = ±§ ln(1 + t3) + y2; —[1 — exp(—3ó02/2)]1/3 < t < æ
17. y — 3 if Ó0 > 0; y = 0 if Ó0 = °; y ——æ if Ó0 < 0
18. y — —æ if Ó0 < 0; y — 0 if Ó0 > 0 19. y — 0 if Ó0 < 9; y-
20. y — —æ if y0 < yc ~ —0.019; otherwise y is asymptotic to Vt — 1
21. (a) No (b) Yes; set t0 = 1/2 in Eq. (19) in text.
(c) |y| < (4/3)3/2 = 1.5396
22. (a) y1(t) is a solution for t > 2; y2(t) is a solution for all t
(b) f is not continuous at (2, —1)
16. y
æ if Ó0 > 9
0
684
Answers to Problems
See SSM for detailed solutions to 23a, 24, 25, 27ab
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