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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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28, 29, 32
3, 6, 7c, 9
11, 14, 16bc, 17a, 18abc
20bd, 21abd, 24abc, 26ab
3
5, 7, 9, 12, 14
15
1 1 rt
26 (a)yi(t) = —Ó() = /, Ks)g{s) ds
1 Mt) MO Jto
28. y =±[5t/(2 + 5ct5)]1/2 29. y = r/(k + cre-rt)
30. y = ±[e/(a + cee-2et)]1/2 f /Ã , t D1/2
31. y
. y = ± „»/ 2/ ^(s) ds + c
, where ä(³) = exp(2T sin t + 2Tt)
32. y = 1 (1 - e-2t) forO < t < 1; y = 1 (e2 - 1)e-2t for t > 1
33. y = e-2t forO < t < 1; y = e-(t +1) for t > 1
Section 2.5, page 84
1. y = 0 is unstable
2. y = —a/b is asymptotically stable, y = 0 is unstable
3. y = 1 is asymptotically stable, y = 0 and y = 2 are unstable
4. y = 0 is unstable 5. y = 0 is asymptotically stable
6. y = 0 is asymptotically stable
7. (c) y = [yo + (1 - yo)kt]/[1 + (1 - yo)kt]
8. y = 1 is semistable
9. y = — 1 is asymptotically stable, y = 0 is semistable, y = 1 is unstable
10. y = —1 and y = 1 are asymptotically stable, y = 0 is unstable
11. y = 0 is asymptotically stable, y = b2/a2 is unstable
12. y = 2 is asymptotically stable, y = 0 is semistable, y = —2 is unstable
13. y = 0 and y = 1 are semistable
a) t = (1/r) ln4; 55.452 years (b) T = (1/r) ln[?(1 - a)/(1 - 0)a];
75.78 years
a) y = 0 is unstable, y = ^ is asymptotically stable
b) Concave up for 0 < y < K/ e, concave down for K/e < y < K
a) y = Kexp{[ln(y0/K)]e-rt} (b) y(2) = 0.7153K = 57.6 x 106 kg
c) t = 2.215 years
b) (h/a)^k/àï; yes 19. (b) k2/2g(aa)2
c) k/à < ï a2
c) Y = ?y2 = KE[1 — (E/r)] (d) Ym = Kr/4 for E = r/2
a) Ó12 = KT[1 T V1 - (4h/rK)]/2
a) y = 0 is unstable, y = 1 is asymptotically stable
b) y = Óî/[.Óî + (1 - Óî)ºãÑ‘²]
a) Ó = y{)e~et (b) x = xo exp[-ayo(1 - e-et)/?] (c) xo exp(-ayo/e)
b) z = 1/[v + (1 - v)eet] (c) 0.0927
pq[ea(q—p)t - 1]
a) lim x(t) = min(p, q); x(t) =
15.
16.
17.
18.
20.
21.
22.
23.
24.
26.
b) lim x(t) = p; x(t) =
2
p at
³^æ 4 ' pat + 1
Section 2.6, page 95
qea(q— p)t - p
1. x2 + 3x + y2 - 2y = c 2. Not exact
3. x3 - x2y + 2x + 2Ó3 + 3y = c 4. x2 y2 + 2xy = c
5. ax2 + 2bxy + cy2 = k 6. Not exact
7. ex sin y + 2y cos x = c; also y = 0 8. Not exact
9. exy cos 2x + x2 --- 3 y = c 10. yl
n
x
+
3
x2
1
2
y
II
c
11. Not exact 12. x2 + y2 = c
13. Ó = [x +-s/28 - 3x^]/2, |x| < V28/3
14. y = [x - (24x3 + x2 - 8x - 16)1/2]/4, x> 0.9846
15. b = 3; x2 y2 + 2x3 y = c 16. b = 1; e2xy + x2 = c
685
See SSM for detailed solutions to 19, 22, 23
25, 26, 27, 29, 31
1d, 3a
3bcd, 4d, 6,9
13a, 15ac, 16
17 . J N(x, 7) d7 + f [ M(x, 7) - J Nx (x, 7) dy^ dx
19. x2 + 2 ln 171— 7-2 = c; also 7 = 0 20. ex sin7 + 27cos x = c
21. xy2 — (y2 — 27 + 2)e7 = c 22. x2ex sin7 = c
24. /x(t) = exp JR(t) dt, where t = xy
25. n,(x) = e3x; (3x27 + y3)e3x = c 26. /x(x) = e-x; 7 = cex + 1 + e2x
27. ä(ó) = 7; xy + 7cos 7 — sin7 = c
28. /ë(ó) = e27 / 7; xe2y — ln |y| = c; also 7 = 0
29. ä(ó) = sin7; ex siny + y2 = c 30. /x(y) = y2; x4 + 3xy + y4 = c
31. /x(x, 7) = xy; x37 + 3x2 + y3 = c
Section 2.7, page 103
1. (a) 1.2, 1.39, 1.571, 1.7439
(b) 1.1975, 1.38549, 1.56491, 1.73658
(c) 1.19631, 1.38335, 1.56200, 1.73308
(d) 1.19516, 1.38127, 1.55918, 1.72968
2. (a) 1.1, 1.22, 1.364, 1.5368
(b) 1.105, 1.23205, 1.38578, 1.57179
(c) 1.10775, 1.23873, 1.39793, 1.59144
(d) 1.1107, 1.24591, 1.41106, 1.61277
3. (a) 1.25, 1.54, 1.878, 2.2736
(b) 1.26, 1.5641, 1.92156, 2.34359
(c) 1.26551, 1.57746, 1.94586, 2.38287
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