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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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2, 2), center, stable; (— 1, 1 ), center, stable
a, c) (0, 0), saddle point, unstable; (\f&, 0), spiral point, asymptotically stable;
—\f6, 0), spiral point, asymptotically stable
a, c) (0, 0), saddle point, unstable; (—2, 2), node, unstable;
4, 4), spiral point, asymptotically stable
a, c) (0, 0), spiral point, unstable 15. (a) 4x2 — y2 = c
a) 4X2 + y2 = c 17. (a) (y — 2x)2(x + y) = c
a) arctan(y/x) — ln y7x2 + y2 = c 19. (a) 2X2y — 2xy + y2 = c
a) x2y2 — 3x2y — ly2 = c 21. (a) (y2/2) — cos x = c
a) x2 + y2 — (x4 /12) = c Section 9.3, page 487
1. linear and nonlinear: saddle point, unstable
2. linear and nonlinear: spiral point, asymptotically stable
3. linear: center, stable; nonlinear: spiral point or center, indeterminate
4. linear: improper node, unstable; nonlinear: node or spiral point, unstable
5. (a, b, c) (0, 0); u' = — 2u + 2v, v' = 4u + 4v; r = 1 ħ\/17;
saddle point, unstable
(—2, 2); u'= 4u, v'= 6u + 6v; r = 4, 6; node, unstable
(4, 4); u' = —6u + 6v, v = —8u; r =—3 ħ s/39i; spiral point, asymptotically stable
6. (a, b, c) (0, 0); u'= u, v' = 3v; r = 1, 3; node, unstable
(1, 0); u' = —u — v, v' = 2v; r = —1, 2; saddle point, unstable
(0, 3); u' = — 1 u, v'= — 3u — 3v; r =—1, —3; node, asymptotically stable
(—1, 2); u'= u + v, v'= —2u — 4v; r = (—3 ħ-\/17)/2; saddle point, unstable
7. (a, b, c) (1, 1); u' = —v, v' = 2u — 2v; r = — 1 ħ i; spiral point, asymptotically
stable
(—1, 1); i/ = —v, v' = —2u — 2v; r =—1 ħ \f3; saddle point, unstable
8. (a, b, c) (0, 0); u' = u, v' = 1 v; r = 1, 1; node, unstable
node, asymptotically stable
node, asymptotically stable
(0, 2); ul = - -u, u; = -?u - 2u; ^ -1 -11 1, 25
(1, 0); u! = - -u — V. . v' = -:4 V; r — —1 —11 1 ’ 4!
( 2 - 2 ); u = -2u — 1 / 3 2 V u 8V; r =(
saddle point, unstable
Answers to Problems
721
See SSM for detailed solutions to 10ab
10c
18abc
22ab,23a, 27a 27b, 28ab
3bc
3e
9. (a, b, c) (0, 0); d = — u + v, V = 2u; r = —2, 1; saddle point, unstable
(0, 1); u' = —u — v, v' = 3u; r = (—1 ħ V?T i )/2; spiral point, asymptotically
stable
(—2, —2); d = —5u + 5v, V = —2v; r = —5, —2; node, asymptotically stable
(3, —2); d = 5u + 5v, v'= 3v; r = 5, 3; node, unstable
10. (a, b, c) (0, 0); d = u, v' = v; r = 1, 1; node or spiral point, unstable
(—1, 0); u1 = —u, v' = 2v; r = — 1, 2; saddle point, unstable
11. (a, b, c) (0, 0); u' = 2u + v, v' = u — 2v; r = ħ V5; saddle point, unstable
(—1.1935, —1.4797); u' = -1.2399u — 6.8393v, v'= 2.4797u — 0.80655v;
r =—1.0232 ħ 4.1125i; spiral point, asymptotically stable
12. (a, b, c) (0, ħ2nn), n = 0, 1, 2,... ; u' = v, v' = —u; r = ħi; center or spiral
point, indeterminate
[2, ħ(2n — 1)n ], n = 1, 2, 3,...; u' = — 3v, v' = — u; r = ħV3;
saddle point, unstable
13. (a, b, c) (0, 0); u'= u, v' = v; r = 1, 1; node or spiral point, unstable
(1, 1); u' = u — 2v, v' = —2u + v; r = 3, —1; saddle point, unstable
14. (a, b, c) (1, 1); u' = — u — v, v' = u — 3v; r =—2, —2;
node or spiral point, asymptotically stable
(—1, —1); u = u + v, v' = u — 3v; r =— 1 ħ V5; saddle point, unstable
15. (a, b, c) (0, 0); u' = — 2u — v, v' = u — v; r = (—3 ħ V3 i)/2;
spiral point, asymptotically stable
(—0.33076, 1.0924) and (0.33076, —1.0924); u' = -3.5216u — 0.27735v,
v' = 0.27735u + 2.6895v; r =—3.5092, 2.6771; saddle point, unstable
16. (a, b, c) (0, 0); u' = u + v, v' = —u + v; r = 1 ħ i; spiral point, unstable
19. (b,c) Refer to Table 9.3.1.
21. (a) ? = 4, T = 3.17 (b) ? = 4, T = 3.20, 3.35, 3.63, 4.17
(c) T ^ n as 4 ^ 0 (d) 4 = n
22. (b) vc = 4.00 23. (b) vc = 4.51
28. (a) dx/dt = y, dy/dt = —g(x) — c(x)y
(b) The linear system is dx/dt = y, dy/dt = —g(0)x — c(0)y.
(c) The eigenvalues satisfy r2 + c(0)r + g (0) = 0.
Section 9.4, page 501
1. (b, c) (0, 0); u' = 3u, v' = 2v; r = 2, 2; node, unstable
(0, 2); u' = 1 u, v' = — 3u — 2v; r = 2, —2; saddle point, unstable
o /44/4 4 4
(2, 0); u = — 3u — 3v, v = | v; r = —3, |; saddle point, unstable
(4, 7); u' = — 4u — 2v, v = — 20u — 5 v; r = (—22 ħ V204)/20;
node, asymptotically stable
2. (b, c) (0, 0); u' = 3u, v' = 2v; r = 3, 2; node, unstable
(0, 4); u' = —1 u, v' = —6u — 2v; r = —1, —2; node, asymptotically stable
o / 44/1 14
(2, 0); u =— 3u — 3v, v = —1 v; r =— 1, — 2; node, asymptotically stable
(1, 1); u = —u — 2v, v' = — 2u — 3v; r = (—3 ħVh3)/4;
saddle point, unstable
3. (b, c) (0, 0); u' = 3u, v' = 2v; r = 3, 2; node, unstable
(0, 2); u' = —1 u, v' = — 9u — 2v; r = — 2, —2; node, asymptotically stable
(3, 0); u' = — 3u — 3v, v' = —?v; r = — 3, — 22; node, asymptotically stable
(5, H); u' = -2u — 4v, v' = -§9u — Hv; r = -1.80475, 0.30475;
saddle point, unstable
722
Answers to Problems
See SSM for detailed solutions to 5bc, 6bc
6d
6ef, 8a 8b, 9ab, 12a
12bcd
3bc
3ef
4. (b, c) (0, 0); u' = 3u, v' = 3v; r = 3, 3; node, unstable
(0, 4); u' = 4u, V = — 4v; r = ħ4; saddle point, unstable
(3, 0); u'= —2u — 3v, V = 8v; r =— 2, |; saddle point, unstable
(2, 2); u' = -u — 2v, v'= —16u — 2v; r =-1.18301, —0.31699;
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