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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.
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s n—0
F (s) — - (e—s + 2e — 6e—4s) s
s\ / „2
] 12. F (s) — (1 — e—s )/s2
14. f(t) — 1 u2(t)[et—2 — e—2(t—2)]
16. f (t) — u2(t) sinh2(t — 2)
18.
f (t) — u1(t) + u2(t) — u3(t) — u4(t)
cos t
21. f (t) — 1 e 23. f(t) — 1 et/2u2 (t/2)
--- e---3s), s > 0
e-2ns - e- -(2n+1)s ]
1/s s > 0

]—
1e
(2n+2)s
s(1 + e—s)
s > 0
29. ?{ f(t)} —
31. L{ f(t)} —
1/s
1 + e—s
1 — (1 + s )e—s s 2(1 — e—s)
s > 0
s > 0
30. L{ f (t)} — 32. L{ f (t)} —
s > 0
34. (b) L{p(t)} —
33. (a) L{ f (t)} —s—1(1 — e—s), s > 0
(b) L{g(t)} — s—2(1 — e—s), s > 0
(c) L{h(t)} — s—2(1 — e—s)2, s > 0
Section 6.4, page 321
1. y — 1 — cos t + sin t — un/2(t)(1 — sin t)
2. y — e—* sin t + 2un(t)[1 + e~(t-lt) cos t + e—1f*-:^ sin t]
1 — e—s s(1 + e—s-1 + e—
(1 + s2 )(1 — e—n s) 1- e—s
s2(1 + e—s)
, s > 0 s > 0
— 1 u2n(t)[1 — e—(t—2n) cos t — e—(t—2n) sin t] 1
2 2n
3. ó — 6[1 — u2n(t)](2sint — sin21)
4. y — 6 (2 sin t — sin21) — 1 un(t)(2 sin t + sin21)
5. y — 1 + 2e 2* — e * — u10(t)[1 + 1 e
1 e— 2(t —10)
-(t —10)
705
See SSM for detailed solutions to 8, 10, 16bc
19abd 20,20abc
1,3, 5, 7
10
13a, b
6. ó = e-t - e-2t I u2(t)[2 - e-(t-2) I 1 e-2(t-2)]
7. ó = cos 11 u3n(t)[1 — cos(f — 3n)]
8. ó = h(t) — un/2(t)h(t — n/2), h(t) = 25(—4 + 5t + 4e-t/2cos t — 3e-t/2sin t)
9. 7 = 1 sint + 11 — 1 u6(t)[t — 6 — sin(f — 6)]
10. 7 = h(t) + un(t)h(t — n), h(t) = y7[—4 cos t + sin t + 4e-t/2 cos t + e-t/2 sin t]
11. 7 = ^(tX1 - 4 cos(2t - 2n)] - u3n(t)[i - 1 cos(2t - 6n)]
12. 7 = u1(t)h(t — 1) — u2(t)h(t — 2), h(t) = — 1 + (cos t + cosh t)/2
13. 7 = h(t) — un(t)h(t —n), h(t) = (3 — 4cost + cos2t)/12
14. f(t) = [u^(t)(t - t0) - u^+k(t)(t - t0 - k)](h/k)
15. #(t) = [utg(t)(t - t0) - 2u°^+k(t)(t - t0 - k) + u^+2k(t)(f - t0 - 2k)](h/k)
16. (b) u(t) = 4ku3/2(t)h(t - 2) - 4ku5/2(t)h(t - 5),
h(t) = 1 - (V7/84) e-t/8 sin(^V7 t/8) - 4e-t/8 cos(^v/7 t/8)
(d) k = 2.51 (e) t = 25.6773
17. (a) k = 5
(b) 7 = [u5(t)h(t - 5) - u5+k(t)h(t - 5 - k)]/k, h(t) = 41 - |sin2t
18. (b) fk(t) = [u4-k(t) - u^k(t)]/2k;
7 = [u4-k(t)h(t - 4 + k) - u4+k(t)h(t - 4 - k)]/2k,
h(t) = 4 - 4e-t/6 cos^V^43 t/6) - (v'T43'/572) e-t/6 sin(v/Ò43 t/6)
n
19. (b) 7 = 1 — cos t + 2 ^ (—1)kukn(t)[1 — cos(t — kn)]
k=1
21. (b) 7 = 1 - cos t + Y, (-1)kukn(t)[1 - cos(t - kn)]
k=1
23. (a) 7 = 1 - cos t + 2 ? (-1)ku11 (t)[1 - cos(t - 11k/4)]
k=1
Section 6.5, page 328
1. 7 = e-t cos t + e-t sin t + un(t)e-(t—) sin(t - n)
2. 7 = 1 un(t) sin2(t — n) — 1 u2n(t) sin2(t — 2n)
3. 7 = — e-2t + 1 e-t + u5(t)[-e-2(t-5) + e-(t-5)] + u10(t)[1 + 1 e-2(t-10) - e-(t-10)]
4. 7 = cosh(t) — 20u3(t) sinh(t — 3)
5. 7 = 4 sin t — 1 cos t + 1 e-t cos^2 t + (1 /^2) u3n(t)e~(t-3n) sin\/2(t — 3n)
6. 7 = 2cos2t + 1 u4n(t) sin2(t — 4n)
7. 7 = sin t + u2n(t)sin(t — 2n)
8. 7 = un/4(t) sin2(t - n/4)
9. 7 = un/2(t)[1 - cos(t - n/2)] + 3u3n/2(t) sin(t - 3n/2) - u2n(t)[1 - cos(t - 2n)]
10. 7 = (1/V^) un/6(t) exp[-1 (t - n/6)] sin(V31/4)(t - n/6)
11. 7 = 5 cos t + 5 sin t - 1 e-t cos t - 3e-t sin t + un/2(t)e-(t-7t/2) sin(t - n/2)
12. 7 = u1(t)[sinh(t — 1) — sin(t — 1)]/2
13. (a) -e-T/4S(t - 5 - T), T = 8ë/V45
14. (a) 7 = (4/VT5) u1(t)e-(t-1)/4 sin(vi5/4)(t - 1)
(b) t1 = 2.3613, y1 = 0.71153
(c) 7 = (8^7/21) u1 (t)e-(t-1)/8 sin(3 V7/8) (t - 1); t1 = 2.4569, y1 = 0.83351
(d) t1 = 1 + n/2 = 2.5708, y1 = 1
15. (a) k1 = 2.8108 (b) k1 = 2.3995 (c) k1 = 2
16. (a) <j)(t, k) = [u4-k(t)h(t -4 + k) - u4+k(t)h(t - 4 - k)]/2k, h(t) = 1 - cos t
(b) u4(t) sin(t — 4) (c) Yes
706
See SSM for detailed solutions to 17b
21b
25b
1c
4
8, 13, 15, 17
20
20
17. (b) ó = Y, èêë (t) sin(t — kn)
k=1
20
20
18. (b) ó =?(—1)k+f ukn(t) sin(t — kn) k=1
19. (b) Ó = E ukn/2(t) sin(t — kn/2)
k=1
20
20. (b) ó = ¨ (—1)k+1ukn/2(t) sin(t — kn/2)
k=1
21 (b) Ó = E u(2k— f)n(t) sin[t — (2k — 1)n]
k=1
40
22. (b) ó = ? (—1)k+-u11k/4(t) sin(t — llk/4)
(t—kn)/20 sin[V399(t — kn)/20]
20
23. (b) Ó = Ji?(—^+4^ 399 k=1 15
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