Books
in black and white
Main menu
Share a book About us Home
Books
Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics
Ads

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
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) = , T = 3.17 (b) = , T = 3.20, 3.35, 3.63, 4.17
(c) T ^ n as ^ 0 (d) = 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
^ 'X 'X
(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;
saddle point, unstable
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, ; node, asymptotically stable
(5, -jl); u' = -2u - 4v, v' = -fu - 11 v; r = -1.80475, 0.30475;
saddle point, unstable
722
Answers to Problems
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
/ 1 / " " 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 = , 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 21 = 0: (0, 0), (0, ^,.), (1/a1, 0)
12 21 = 0: (0, 0), and all points on the line a1 x + 1 y = e1
(b) a1e2 21 > 0: (0, 0) is unstable node; (1/1, 0) is saddle point;
(0, ^2/a2) is asymptotically stable node.
a1e2 21 < 0: (0, 0) is unstable node; (0, 2/2) is saddle point;
(1/1, 0) is asymptotically stable node.
(c) (0, 0) is unstable node; points on the line a1 x + 1 y = e1 are stable, nonisolated critical points.
12. (a) (0, 0), (0, 2 + 2), (4, 0), (2, 2)
(b) = 0.75, asymptotically stable node; = 1.25, (unstable) saddle point
(c) vi = 2u 2v, v' = 2u 2v
(d) r = 2 2.; 0 = 1
13. (a) (0, 0), saddle point; (0.15, 0), spiral point if 2 < 1.11, node if 2 > 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