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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
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14, 21, 22, 27
28, 30
1
3
17. y — te‘ — t e‘ + 21 et 18. y — cosh t
19. y — cosV21
20. y — (a2 — 4)—1 [(a2 — 5) cos at + cos 2f]
21. y — 1 (cos t — 2 sin t + 4e( cos t — 2e( sin t)
22. y = 5 (e—r — er cos t + 7er sin t) s 1 - e—n s
23. y = 2e—t + te—'t + 2t2e—t
1 e—s (s + 1)
25. Y (s) = -^-2- ( )
29. 1/(s — a)2 31. n\/s‘
n+1
24. Y (s) — —2----------+ 2
s + 4 s(s2 + 4)
26. Y (s) — (1 — e—s )/s2(s2 + 4)
30. 2b(3s2 — b2)/(s2 + b2)3
32. n!/(s — a)n+1
34. [(s — a)2 — b2]/[(s — a)2 + b2]2
36. (a) Y + s2 Y — s (b) s2 Y" + 2sY'— [s2 + a(a + 1)]Y —-1 Section 6.3, page 314
s2(s2 + 1) s2(s2 + 1)
33. 2b(s — a)/[(s — a)2 + b2]2
7
9
11
13
F (s) —
F (s) — 2e—s/s3
n s
F (s) — e—s (s2 + 2)/s3
2 — 2 s2 s2
-(1 + n s)
F(s) — s—2[(1 — s)e—2s — (1 + s) e—3s f (t) — t3e2t
15. f (t) — 2u2(t)et—2cos(t — 2)
17. f (t) — u 1 (t)e2(t—1) cosh(t — 1)
20. f(t) — 2(2t)n
22 24
25. F (s) — s~ ‘(1 — +
26. F(s) — 1[1 — e—s +
s
27. F(s) — - J2(— 1)ne—'ns —
s n—0
f (t) — 6 et/3(e2t/3 — 1)
F(s) — s—1(1 — e—?s), s > 0
10.
12.
14.
16.
18.
21.
23.
1
F (s) — - (e—s + 2e — 6e—4s)
s
—s 2
—2(t—2)
F(s) — (1 — e—s)/s2 f (t) — 3 u2(t)[et—2 — e-f (t) — u2(t) sinh2(t — 2) f (t) — u1(t) + u2(t) — u3(t) — u4(t)
- 1 0-‘/2
cos t
f (t) — 2 e f (t) — 2 et/2u2 (t/2)
e 3s), s > 0
e 2ns e (2n+1)s]
1/s s > 0
1 + e—s ’

1e
(2n+2)s
s(1 + e—s)
s > 0
29. L{ f (t)} —
1/s
1 + e—
s > 0
30.
32.
34.
1 — (1 + s)e^s
31. L{ f(t)}—^-------------—— , s > 0
s2(1 — e s)
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— t sin t + 1 un(t)[1 + e— (t—n) cos t + e— (t—n)
— 1 u2n(t)[1 — e— (t—2n) cos t — e— (t—2n) sin t]
3. y — 6[1 — u2n(t)](2sint— sin21)
4. y — 6(2 sin t — sin21) — 1 u_(t)(2 sin t + sin21)
L{ f (t)} — L{ f (t)} —
s > 0
(b) L{p(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
sint]
5. y — 1 + 2e 2‘— e ‘— ulo(t)[- +1 e
1 e— 2(t —10)
-(t—10)
e
e
s
e
Answers to Problems
705
See SSM for detailed solutions to 8, 10, 16bc
19abd 20, 20abc
1, 3, 5, 7
10
13a, b
6. / = e-t - e-2t + u2(t)[2 - e-(t-2) + Ie-2(t-2)]
7. y = cos t + u3n(t)[1 - cos(f - 3n)]
8. y = h(t) - un/2(t)h(t - n/2), h(t) = 25(-4 + 5t + 4e-t/2cos t - 3e-t/2sin t)
9. y = 1 sin t + 11 - Iu6(t)[t - 6 - sin (f - 6)]
10. y = h(t) + un(t)h(t - n), h(t) = i7[-4 cos t + sin t + 4e-t/2 cos t + e-t/2 sin t]
11. y = un(.t)[\ - 4 cos(2t - 2n)] - u3n(t)[4 - 4 cos(2t - 6n)]
12. y = u1(t)h(t - 1) - u2(t)h(t - 2), h(t) = -1 + (cos t + cosh t)/2
13. y = 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. g(t) = [u^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 - (77/84) e-t/8 sin(3T71/8) - 4e-t/8 cos(3V71/8)
(d) k = 2.51 (e) t = 25.6773
17. (a) k = 5
(b) y = [u5(t)h(t - 5) - u5+k(t)h(t - 5 - k)]/k, h(t) = 41 - |sin2t
18. (b) fk(t) = [u4-k(t) - U4+k(t)]/2k;
y = [u4-k(t)h(t - 4 + k) - u4+k(t)h(t - 4 - k)]/2k,
h(t) = 4 - 4e-t/6 cos^vT43 t/6) - (7143/572) e-t/6 sin(7?43 t/6)
n
19. (b) y = 1 - cos t + 2 ^ (- 1)kukn(t)[1 - cos(t - kn)]
k=1
21. (b) y = 1 - cos t + ? (-1)kukn(t)[1 - cos(t - kn)]
k=1
23. (a) y = 1 - cos t + 2 ? (-1)ku (t)[1 - cos(t - 11k/4)]
k=1
Section 6.5, page 328
1. y = e-t cos t + e-t sin t + un(t)e-(t-n) sin(t - n)
2. y = 1 un(t) sin2(t - n) - 1 u2n(t) sin2(t - 2n)
3. y = -1 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. y = cosh(t) - 20u3(t) sinh(t - 3)
5. y = 4 sin t - 1 cos t + 1 e-t cos 72 t + (1 /72) u3n(t)e-(t-3n) sin72(t - 3n)
6. y = icos2t + 1 u4n(t) sin2(t - 4n)
7. y = sin t + u2n(t)sin(t - 2n)
8. y = un/4(t) sin2(t - n/4)
9. y = un/2(t)[1 - cos(t - n/2)] + 3u3n/2(t) sin(t - 3n/2) - u2n(t)[1 - cos(t - 2n)]
10. y = (1/731) un/6(t) exp[-1 (t - n/6)] sin(T3?/4)(t - n/6)
11. y = 5 cos t + 5 sin t - 1 e-t cos t - 3e-t sin t + un/2(t)e-(t-n/2) sin(t - n/2)
12. y = u1(t)[sinh(t - 1) - sin(t - 1)]/2
13. (a) -e-T/48(t - 5 - T), T = 8n/7?5
14. (a) y = (4/715) u1(t)e-(t-1)/4 sin^715/4)(t - 1)
(b) t1 = 2.3613, y1 = 0.71153
(c) y = (877/21) u1 (t)e-(t-1)/8 sin(^77/8) (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
Answers to Problems
See SSM for detailed solutions to 17b
21b
25b
1c
4
3, 13, 15, 17
20
20
17. (b) y = E ukn (t) sin(t - kn)
k=1
20
20
18. (b) y =? (-1)k-1 ukn(t) sin(t - kn)
k=1
19. (b) y = E ukn/2(t) sin(t - kn/2)
k=1
20
20. (b) y = E (-1)k+l ukn/2(t) sin(t - kn/2)
k=1
21. (b) y = E u(2k—1)n(t) sin[t - (2k - 1)n]
k=1
40
22. (b) y = E (-1)k+1u11k/4(t) sin(t - 11k/4)
(t-kn)/20 sin[V399(t - kn)/20]
20
23. (b)y =72^^(-1)k+1«kn(t)e
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