# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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2. (b, c) (0, 0); u' = u,v = o v; r = 1, 4; saddle point, unstable

(1, 2); u' = 4v, v' = u; r = ±1 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, f; saddle point, unstable

(2, f); u' = 4u 1 v, v = 3 u; r = (1 ± ė/Či)/8; spiral point,

asymptotically stable

4. (b, c) (0, 0); u' = 9 u,v' = v; r = |, 1; saddle point, unstable

(8, 0); u' = 8u 16v,v' = 1 v; r = |, 8; saddle point, unstable

(1, 4); u' = u 2v, v = 4u; r = (1 ± 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 = 2 v; r = 1, 3; node, asymptotically stable

(1, 0); u! = 4 u Žv, v = v; r = 1, 3; saddle point, unstable

(2, 0); il = 3u 5v,v'= 2v; r = 3, 2; saddle point, unstable

(3, 3); vl = 3 u Žv, v1 = 5 u; r = (3 ± \/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 is a min., P is a max.

t = T/2, 3T/2,... : H is a min., dP/dt is a min.

t = 3T/4, 7T/4,... : dH/dt is a max., P is a min.

7. (a) Jcą/Ja y (b) Jb

(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 ą HP ā H, dP/dt = cP + y HP & P, where ā and & are constants of proportionality. New center is H = (c + &)/y > c/y and P = (a ā)/ą < a/ą, so equilibrium value of prey is increased and equilibrium value of predator is decreased!

13. Let A = a/ą c/y > 0. The critical points are (0, 0), (a/a, 0), and (c/y, a A/ą), where (0, 0) is a saddle point, (a/a, 0) is a saddle point, and (c/y, a A/ą) 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/ą) as t ^ ę.

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 = v/0, 9 = t + t0, unstable periodic solution

14. (a) /ė = 0.2, T = 6.29; ä = 1, T = 6.66; ä = 5, T = 11.60

15. (a) x ^ y = x + äó - ėó3/3

(b) 0 < ä < 2, unstable spiral point; ä > 2, unstable node

(c) A = 2.16, T = 6.65

(d) ä = 0.2, A = 1.99, T = 6.31; ä = 0.5, A = 2.03, T = 6.39;

ä = 2, A = 2.60, T = 7.65; ä = 5, A = 4.36, T = 11.60

16. (a) k = 0, (1.1994, -0.62426); k = 0.5, (0.80485, -0.13106)

(b) k0 = 0.3465 (c) T = 12.54 (d) k1 = 0.3369

Section 9.8, page 538

1ab

1. (b) X = ė1, ?(1) = (0, 0, 1)T; X = A.2, g(2) = (20, 9 - V81 + 40r, 0)T;

X = X3, ?(3) = (20, 9 + J81 + 40r, 0)T

724

See SSM for detailed solutions to 1c, 2abc, 3b

3c, 4, 5ab

CHAPTER 10

2, 3, 7, 11, 15

3, 5, 7, 10

13ab, 15ab

15a, 21 abed, 25

Answers to Problems

(c) X1 = -2.6667, ?(1) = (0, 0, 1)T; X2 = -22.8277, g(2) = (20, -25.6554, 0)T; X3 = 11.8277, g(3) = (20, 43.6554, 0)T

2. (c) X1 = -13.8546; X2,X3 = 0.0939556 ± 10.1945Ó

5. (a) dV/dt = -2a[rx2 + y2 + b(z - r)2 - br2]

Section 10.1, page 547

2. y = (cot \fln cosV2x + sin v/2x)/v/2

1. y =- sin x

3. y = 0 for all ³; y = c2 sin x if sin L = 0.

4. y = - tan L cos x + sin x if cos L = 0; no solution if cos L = 0.

5. No solution.

6. y = (-n sinV^x + xsinv/2n)/2 sinv/2n

7. No solution. 8. y = c2sin2x + 1 sin x

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