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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
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(b) a = 2
24. 2() = f 2
26. y2(f) = fef
28. y2(x) = x
30. y2(x) = x1/2 cos x
ds
35. y2(f) = cos f2 37. y2(x) = x1/2 cos x 41. y= c1 f2 + c2f2ln f
Section 3.6, page 178
1. = c1e3f + c2e f e2f
e f cos2f + c2e f sin2f + 17sin2f 12cos2f
e3f + c2ef + 16 fe f + 3 f2 e f
+ c2e 2f + 2f 2 sin2f 2 cos2f
cos 3f + c2 sin3f + ^(9f2 6f + 1)e3f + 3
e t + c2te t + t 2e t
e f + c2e f/2 + f2 6f + 14 -6 sin f 10 cos f cos f + c2 sin f 3 f cos2f sm2f cos w0f + c2 sin0f + (2 w2)1 cos wf cos rn0f + c2 sinrn0f + (1 /2w0)f sinw0f
e1/2 cos(\/T5 t/2) + c2ef/2 sin(vT5 f/2) + 1 ef 4e f
2. y = c1
3. y = c1
4. y = c1
5. y = c1
6. y = c1
7. y = c1
8. y = c1
9. u = c 10. u = c
. = c 12. = c1
14. = sin2f 19 cos2f + 4 f2 1 + |ef 15. = 4fef 3ef + 6fV + 4
4'
e + c, e2f + 1 fe2f + 1 e
2t
13. = ef 1 e2ff 2
1 +3J
17. = 2cos2f 1sin2f 3fcos2f
16. = 3' + 3ef 3e2f fe2'
18. = e f cos2f + 1 e f sin2f + fe f sin2f
19. (a) Y(f) = f(V4 + A
+ D sin3f + E cos 3f
(b) A0 = 2/15, A1 = 2/9, A2 = 8/27, A3 = 8/27, A4 = 16/81, B0 = 1/9,
B1 = -1/9, B2 = 2/27, D = 1/18, E = 1/18
20. (a) Y(f) = A0f + A1 + f(B0f + B1) sin f + f(D0f + D1) cos f
(b) A0 = 1, A1 = 0, B0 = 0, B1 = 1/4, D0 = 1/4, D1 = 0
21. (a) Y(f) = ef(Acos2f + Bsin2f) + (D0f + D1)e2f sin f + (E0f + E1)e2f cos f
(b) A = 1/20, B = 3/20, D0 = 3/2, D1 = 5, E0 = 3/2, E1 = 1/2
22. (a) Y(f) = Ae f + f(B0f2 + B1f + B2)e f cos f + f(D0f2 + D1f + D2)ef sin f
(b) A = 3, B0 = 2/3, B1 =0, B2 = 1, D0 = 0, D1 = 1, D2 = 1
Answers to Problems
691
See SSM for detailed solutions to 28
30, 33 2
5, 11, 14
18, 22 25, 29
23. (a) (t) = A^ + A11 + A2 + t2(B0t + B1)e2t + (D0t + ) sin2t + (E0t + E1) cos2t
(b) A0 = 1/2, A1 = 1, A2 = 3/4, B0 = 2/3, B1 = 0, D0 = 0, D1 = -1/16,
E0 = 1 /8, E1 = 1/16
24. (a) (t) = t(A0t2 + A1t + A2) sin2t + t(B0t2 + B1t + B2) cos2t
(b) A0 = 0, A1 = 13/16, A2 = 7/4, B0 = -1/12, B1 =0, B2 = 13/32
25. (a) (t) = (A0f2 + A1t + A2)e sin2t +(B0t2 + B1t + B2)et cos2t
+ e-t (D cos t + E sin t) + Fet
(b) A0 = 1/52, A1 = 10/169, A2 = -1233/35152, B0 = -5/52, B1 = 73/676,
B2 = -4105/35152, D =-3/2, E = 3/2, F = 2/3
26. (a) (t) = t(A0t + A1)e-t cos2t + t(B0t + B1)e-t sin2t + (D0t + D1)e-2t cos t + (E0t + E1)e-2t sin t
(b) A0 = 0, A1 = 3/16, B0 = 3/8, B1 = 0, D0 = -2/5, D1 = -7/25, E0 = 1/5,
E1 = 10/25 1 0 1 0 1 0
27. y = c1 cos kt + c2 sin kt + ^ [am/(k2 m2n 2)] sin mn t
m=1
28 _{t, 0 < t < n
. y j-(1 + n/2) sin t (n/2) cos t + (n/2)en-f, t >n
1 - 47; e-f sin2t - 1 e-f cos2t, 0 < t < n/2
5 10 5
-1 (1 + en/2)e-t cos2t - ^ (1 + en/2)e^t sin21, t > n/2
30. No 33. y = c1e4t + c2e-t - 2e2^
Section 3.7, page 183
1. (t) = e' 2. (t) = -3 te-t
3. (0 = t2e-t 4. (') = 2t2et/2
5. y = c1 cos t + c2 sin t - (cos t) ln(tan t + sec t)
cos 31 + c2 sin 31 + (sin 31) ln(tan 31 + sec 31) - 1
6. y = c1
7. y = c1
8. y = c1
9. y = c1
10. y = c
11. y = c1
12. y = c1
2t + c2te 2t - e 2t ln l
cos2t + c2 sin2f + 3 (sin2t) lnsin2t - 31 cos2f cos(t/2) + c2 sin(t/2) + t sin(f/2) + 2[lncos(t/2)] cos(t/2) ef + c2tef - 2ef ln(1 + f2) + tef arctan t
e2f + c2e3f + J [e3(f-s) - e2(f-s)] g(s) ds cos2t + c2sin2t + " [sin2(t - s)] g(s) ds
13. (t) = 1 + f2 ln t 14. (t) = 2f2
15. () = 1 (t - 1)e2f 16. () = -^(2t - 1)e-f
17. (x) = ix2(lnx)3 18. (x) = -2x1/2 cosx
7 xef - tex 2 .
19. (x) = J (1 t)2et g(t) dt 20. (x) = x-1/2J t-3/2sin(x - t)g(t) dt
23. (b) y = y0cos t + yd sin t + / sin(f s)g(s) ds
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