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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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-1,-1), spiral point, asymptotically stable; (-2, 0), saddle point, unstable
a, c) (0, 0), saddle point, unstable; (0, 1), saddle point, unstable;
2, 2), center, stable; (-1, 1), center, stable
a, c) (0, 0), saddle point, unstable; (V6, 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 - 71 = c
a) 4x2 + y2 = c __________________ 17. (a) (7 — 2x)2(x + 7) = c
a) arctan(7/x) — ln y7x2 + y2 = c 19. (a) 2x27 - 2x7 + y2 = c
a) x2y2 - 3x27 - 2ำo = c 21. (a) (yo/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 ฑ\fH;
saddle point, unstable
(-2, 2); U = 4u, v' = 6u + 6v; r = 4, 6; node, unstable
(4, 4); u' = —6u + 6v, v' = —8u; r =-3 ฑ 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 ฑ\/17)/2; saddle point, unstable
7. (a, b, c) (1, 1); u' = —v, v' = 2u — 2v; r = — 1 ฑ i; spiral point, asymptotically
stable
(—1, 1); 1/ = —v, v = —2u — 2v; r =—1 ฑ \/3; 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); uI = -u, v v = u - 2v; r--1 -1
1 -, 2’
(1, 0); uI = -u --- v. . v' = -i v; r --- ---1 ---
1, 4!
(2,2); uI = -2u - 13 8v; r =(-
2v, v = --u
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); d = —u — v, v' = 3u; r = (-1 ฑ \/รา 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); u' = — u, v' = 2v; r = — 1, 2; saddle point, unstable
11. (a, b, c) (0, 0); u' = 2u + v, v' = u — 2v; r = ฑ\/5; saddle point, unstable
(-1.1935, -1.4797); u' = -1.2399u - 6.8393v, v'= 2.4797u - 0.80655v;
r =—1.0232 ฑ 4.1125ฒ; spiral point, asymptotically stable
12. (a, b, c) (0, ฑ2nn), n = 0, 1, 2,...; u' = v, v' = —u; r = ฑi; center or spiral
point, indeterminate
[2, ฑ(2n — 1)n ], n = 1, 2, 3,...; u' = — 3v, v' = — u; r = ฑ\/3;
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 ฑ \/5; saddle point, unstable
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