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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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node, asymptotically stable
5. (b, c) (0, 0); u' = u, v' = 3v; r = 1, |; node, unstable
(0, 2); u =— 2u, v =— 3u — 3 v; r =— 2, — 2; node, asymptotically stable
(1, 0); u' = — u — v, v' = 1 v; r =— 1, 1; saddle point, unstable
6. (b, c) (0, 0); u' = u, v' = 5v; r = 1, 5; node, unstable
(0, 5); u' = Hu, v' = 22u — | v; r = H, — 5; saddle point, unstable
(1, 0); u' = — u + 1 v, v' = 22 v; r = — 1, 141; saddle point, unstable
(2, 2); u = —2u + v, v' = 2u — 3v; r = (—5 ą\/3)/2;
node, asymptotically stable
8. (a) Critical points are x = 0, y = 0; x = e1 /a1, y = 0; x = 0, y = ?2/a2.
x ^ 0, y ^ ?2/a2 as f ^to; the redear survive.
(b) Same as part (a) except x ^ ?1/a1, y ^ 0 as f ^to; the bluegill survive.
9. (a) X = (B — y1 R)/(1 — y1y2), Y = (R — y2 B)/(1 — y1y2)
(b) X is reduced, Y is increased; yes, if B becomes less than y1 R, then x ^ 0 and y ^ R
as f ^ to.
10. (a) a1e2 — a2e1 = 0: (0, 0), (0, ?2/a2), (?1/a1, 0)
a1?2 — a2?1 = 0: (0, 0), and all points on the line a1 x + a1 y = e1
(b) a1e2 — a2e1 > 0: (0, 0) is unstable node; (?1/a1, 0) is saddle point;
(0, ?2/a2) is asymptotically stable node.
a1e2 — a2?1 < 0: (0, 0) is unstable node; (0, ?2/a2) is saddle point;
(?1/a1, 0) is asymptotically stable node.
(c) (0, 0) is unstable node; points on the line a1 x + a1 y = ?1 are stable, nonisolated critical points.
12. (a) (0, 0), (0, 2 + 2a), (4, 0), (2, 2)
(b) a = 0.75, asymptotically stable node; a = 1.25, (unstable) saddle point
(c) vi = —2u — 2v, v' = —2au — 2v
(d) r = —2 ą 2^a; a0 = 1
13. (a) (0, 0), saddle point; (0.15, 0), spiral point if y2 < 1.11, node if y2 > 1.11;
( 2 , 0 ), saddle point
(c) y = 1.20
Section 9.5, page 509
1. (b, c) (0, 0); u' = 3 u, v' = — 1 v; r = 2, — 1; saddle point, unstable
(2, 3); u' = — 4v, v' = 3u; r = ą\/3i/2; center or spiral point, indeterminate
2. (b, c) (0, 0); u' = u,v = — 1 v; r = 1, — 4; saddle point, unstable
( 1, 2); u' = —44v, v' = u; r = ą1 i; center or spiral point, indeterminate
3. (b, c) (0, 0); u' = u,v = — 4 v; r = 1, — 1; saddle point, unstable
(2, 0); u' = — u — v,v'= 3 v; r =— 1, 3; saddle point, unstable
(2, 3); u' = — 4u — 1 v, v' = 3 u; r = (—1 ą v2!!i)/8; spiral point,
asymptotically stable
4. (b, c) (0, 0); u' = 9 u,v' = —v; r = |, —1; saddle point, unstable
(8, 0); u' = — 88u — 26v,v' = 1 v; r = —|, 8; saddle point, unstable
(1, 4); u = —u — 2v, v' = 4u; r = (—1 ą V0.5)/2; node, asymptotically stable
Answers to Problems
723
See SSM for detailed solutions to 7abc, 11,13
1, 2, 4, 7, 8a
8b, 9, 11, 13, 16a
5. (b, c) (0, 0); u' = — u, v' = — |v; r = —1, — node, asymptotically stable
( 1, 0); u' = 4 u — 20v, V = —v; r =— 1, 3; saddle point, unstable
(2, 0); u' = — 3u — 5v,v'= 2v; r =— 3, 2; saddle point, unstable
(3, 3); u; = —3 u — 20v, v' = 5 u; r = (—3 ą \/39 i)/8; spiral point,
asymptotically stable
6. t = 0, T, 2T,... : His a max., dP/dt is a max.
t = T/4, 5T/4,... : dH/dt isamin., P is a max.
t = T/2, 3T/2,... : H isamin., dP/dt isamin.
t = 3T/4, 7T/4,... : dH/dt is a max., P isamin.
7. (a) Vca/Ja y (b) V3
(d) The ratio of prey amplitude to predator amplitude increases very slowly as the initial point moves away from the equilibrium point.
8. (a) 4n/\/3 = 7.2552
(c) The period increases slowly as the initial point moves away from the equilibrium point.
9. (a) T = 6.5 (b) T = 3.7, T = 11.5 (c) T = 3.8, T = 11.1
11. Trap foxes in half-cycle when dP/dt > 0, trap rabbits in half-cycle when dH/dt > 0, trap rabbits and foxes in quarter-cycle when dP/dt > 0 and dH/dt > 0, trap neither in quarter-cycle when dP/dt < 0 and dH/dt < 0.
12. dH/dt = aH — a HP — f H, dP/dt = —cP + y HP — & P, where f and & are constants of proportionality. New center is H = (c + &)/y > c/y and P = (a — f)/a < a/a, so equilibrium value of prey is increased and equilibrium value of predator is decreased!
13. Let A = a/a — c/y > 0. The critical points are (0, 0), (a/a, 0), and (c/y, a A/a), where (0, 0) is a saddle point, (a/a, 0) is a saddle point, and (c/y, a A/a) is an asymptotically stable node if (ca/y)2 — 4ca A > 0 or an asymptotically stable spiral point if (ca/y)2 — 4ca A < 0. (H, P) ^ (c/y, a A/a) as t ^ to.
Section 9.7, page 530
1. r = 1,9 = t + t0, stable limit cycle
2. r = 1,9 = —t + t0, semistable limit cycle
3. r = 1,9 = t + t0, stable limit cycle; r = 3, 9 = t + t0, unstable periodic solution
4. r = 1, 9 = —t + t0, unstable periodic solution; r = 2, 9 = —t + t0, stable limit cycle
5. r = 2n — 1,9 = t + t0, n = 1, 2, 3,..., stable limit cycle;
r = 2n, 9 = t + t0, n = 1, 2, 3,..., unstable periodic solution
6. r = 2, 9 = — t + t0, semistable limit cycle;
r = 3, 9 = —t + t0, unstable periodic solution
8. (a) Counterclockwise
(b) r = 1, 9 = t + f0, stable limit cycle; r = 2, 9 = t + f0, semistable limit cycle; r = 3, 9 = t + t0, unstable periodic solution
9. r = \fl, 9 = —t + t0, unstable periodic solution
14. (a) = 0.2, T = 6.29; ^ = 1, T = 6.66; /u = 5, T = 11.60
15. (a) x = v y = —x + uy — uy3/3
(b) 0 < u < 2, unstable spiral point; u > 2, unstable node
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