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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
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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
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
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