# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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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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