Books in black and white
 Books Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics

# 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.
Previous << 1 .. 389 390 391 392 393 394 < 395 > 396 397 398 399 400 401 .. 609 >> Next

15. (a, b, c) (0, 0); u' = — 2u — v, v' = u — v; r = (—3 ± V3i)/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 ± i; spiral point, unstable
19. (b,c) Refer to Table 9.3.1.
21. (a) ß = Ë, T = 3.17 (b) ß = Ë, T = 3.20, 3.35, 3.63, 4.17
(c) T ^ n as Ë ^ 0 (d) Ë = 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
^ ³ 'X 'X ³ Ï
(2, 0); u =— 3u — 3v, v = 7 v; r = — 3, 7; saddle point, unstable
(4, 7); u' = — ^u — 2v, v = — 20u — 5 v; r = (—22 ± 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
(2, 0); u =— 3u — 3v, v =— 1 v; r =—1, — 2; node, asymptotically stable
(1, 1); u = —u — 2v, v = — 2u — 2v; r = (—3 ±\/¿3)/4;
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' = —^u — 3v, v' = —11 v; r = — 3, — Ö; node, asymptotically stable
(5, -jl); u' = -2u - 4v, v' = -f§u - 11 v; r = -1.80475, 0.30475;
722
See SSM for detailed solutions to 5bc, 6bc
6d
6ef, 8a 8b, 9ab, 12a
12 bed
3bc
3ef
4. (b, c) (0, 0); u' = f u, v' = f v; r = f, f; node, unstable
(0, 4); u' = 4u, v' = — 4v; r = ±|; saddle point, unstable
(3, 0); u1 = — 2u — 3v, v = 8v; r =— 2, §; saddle point, unstable
(2, 2); u' = -u - 2v, v; = —16u - 1 v; r =-1.18301, -0.31699;
node, asymptotically stable
5. (b, c) (0, 0); u' = u, v' = fv; r = 1, f; node, unstable
¦Ý / 1 / "Ý "Ý 1*3
(0, 2); u = —2u, v =— fu — 3 v; r =—22, — 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' = 12u — 5 v; r = Ö, — 5; saddle point, unstable
(1, 0); u' = — u + 1 v, v = ^v; r = — 1, 1°; saddle point, unstable
(2, 2); u = —2u + v, v' = 2u — 3v; r = (—5 ±\/3)/2;
node, asymptotically stable
8. (a) Critical points are x = 0, y = 0; x = e1 /a1, y = 0; x = 0, y = e2/a2.
x — 0, y — ^2/a2 as t — to; the redear survive.
(b) Same as part (a) except x — e1/ff1, y — 0 as t — to; the bluegill survive.
9. (a) X = (B - y1 R)/(1 - y1Y2), Y = (R - y2B)/(1 - y1y2)
(b) X is reduced, Y is increased; yes, if B becomes less than y1 R, then x ^ 0 and y ^ R
as t —> to.
10. (a) a1e2 — à2å1 = 0: (0, 0), (0, é^â,.), (ˆ1/a1, 0)
à1å2 — à2å1 = 0: (0, 0), and all points on the line a1 x + à1 y = e1
(b) a1e2 — à2å1 > 0: (0, 0) is unstable node; (é1/â1, 0) is saddle point;
(0, ^2/a2) is asymptotically stable node.
a1e2 — à2å1 < 0: (0, 0) is unstable node; (0, º2/â2) is saddle point;
(é1/â1, 0) is asymptotically stable node.
(c) (0, 0) is unstable node; points on the line a1 x + à1 y = e1 are stable, nonisolated critical points.
12. (a) (0, 0), (0, 2 + 2à), (4, 0), (2, 2)
(b) à = 0.75, asymptotically stable node; à = 1.25, (unstable) saddle point
(c) vi = —2u — 2v, v' = —2àu — 2v
(d) r = —2 ± 2.à; à0 = 1
13. (a) (0, 0), saddle point; (0.15, 0), spiral point if ó2 < 1.11, node if ó2 > 1.11;
( 2 , 0 ), saddle point
(c) y = 1.20
Section 9.5, page 509
1. (b, c) (0, 0); u' = f u, v' = — 1 v; r = 2, — 1; saddle point, unstable
(2, 3); u' = — 4v, v = 3u; r = ±\ff i/2; center or spiral point, indeterminate
Previous << 1 .. 389 390 391 392 393 394 < 395 > 396 397 398 399 400 401 .. 609 >> Next