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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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Answers to Problems
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) y = (c/1) + (3 cos2t)/4t + (3 sin2t)/2; y is asymptotic to (3 sin2t)/2 as t ? (c) y = ce2t — 3e(; y ^to or —to as t ^to (c) y = (c — tcos t + sin t)/12; y ^ 0 as t ^to
2 -4 ,2
(c) y = t2e 1 + ce 1 ; y ^ 0 as t ^ to
(c) y = (arctan t + c)/(1 + t2)2; y ^ 0 as t ^ to
(c) y = ce— t/2 + 3t— 6; y is asymptotic to 3t — 6 as t ^to
(c) y = — te— ( + ct; y ^ to, 0, or — to as t ^to
(c) y = ce— 1 + sin2t — 2 cos2t; y is asymptotic to sin2t — 2 cos2t as t ^to
(c) y = ce— t/2 + 3t2 — 12t + 24; y is asymptotic to 3t2 — 12t + 24 as t ^ to
y = 3er + 2(t- 1)e2t
y = (3t4 - 4t3 + 6t2 + 1)/12t2
y = (t + 2)e2t
y =-(1 + t)e-‘/t\ t = 0
(b) y = —4 cos t + 5 sin t + (a + 4 )et/2;
5 1 1 5
(c) y oscillates for a = a,
(b) y = -3et/3 + (a + 3)ef/2;
(c) y ^ —to for a = a,
(b) y = fe-f + (ea - 1)e-f/1;
(c) y ^ ) as f ^ ) for a = a,
14.
16.
18.
20.
a0=
= (t2 - 1)
e 2t /2
2
y
y = (sin t)/t2
y = t 2[(n2/4) - 1 - tcos t + sin t] y = (t - 1 + 2e-t)/1, t = 0
a3
a0 = 1 /e
24. (b) y =-cos f/12 + n2a/4t2; a0 = 4/n2 (c) y ^ 2 as t ^ 0 for a = a0 (t, y) = (1.364312, 0.820082) 26.
(a) y = 12 + 65 cos2t + 65 sin2t - 76858e-t/4;
65 , i 65 ?
(b) f = 1).065778
y, = -5/2
y) = -16/3; y TOas f -
See Problem 2.
See Problem 4.
y, = -1.642876 y oscillates about 12 as f ?
to for y0 = —16/3
Section 2.2, page 45
1.
2.
3.
4.
5.
6.
7.
8. 9.
11.
13.
15.
17.
19.
3 y2 - 2X3 = C; y = )
3 y2 - 2ln|1 + x31 = c; x = -1, y = )
y-1 + cos x = c if y = ); also y = ); everywhere
3y + y2 - x3 + x = c; y =-3/2
2 tan 2y - 2x - sin2x = c if cos2y = ); also y = ±(2n + 1)n/4 for any integer n; everywhere
y = sin[ln |x| + c] if x = ) and |y| < 1; also y =±1
y2 - x2 + 2(ey - e-x) = c; y + ey = )
3 y + y3 - x3 = c; everywhere (a) y = 1/(x2 - x - 6)
(c) -2 < x < 3
(a) y = [2(1 - x)ex - 1]1/2
(c) -1.68 < x < ).77 approximately
(a) y = — [2ln(1 + x2) + 4]1/2
(c) -TO < x < TO
(a) y = - + 5\/4x2 - 15
(c) x > 2VT5 _______________
(a) y = 5/2 - V*3—^*+13/4
(c) — 1.4445 < x < 4.6297 approximately
(a) y = [n - arcsin(3 cos2 x)]/3
(c) |x - n/2| < 0.6155
10.
12
-1/2
(a) y = - V2x - 2X2 + 4 (c) -1 < x < 2 (a) r = 2/(1 - 2 ln0)
(c) 0 <9 < */e____________
14. (a) y = [3 - 2^1 + x2]"
(c) |x| < 2 Vs 16. (a) y = -(x2 + 1)/2 (c) -to < x < to (a) y = -4 + 2^65 - 8ex - 8e (c) |x| < 2.0794 approximately (a) y = [|(arcsinx)2]1/3
(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 + ^2x- 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 yo > 0; y = 0 if yo = 0; y ^-ro 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 + 5555 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(ert - 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 12 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, 31 bdef
1, 4, 8, 11, 13, 17
22abc
19. (a) x = 0.04[1 - exp(— t/12,000)] (b) t = 36min
0 — k — (P/r)]e-rt/ V; lim c = k + (P/r)
(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
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