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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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¦ + anf
(a) a0[n(n — 1)(n — 2) ••• 1] + a1[n(n — 1) ••• 2]f + •
(b) (a0rn + a1rn—1 + ••• + an)erf
(c) ef, e—f, e2f, e—2f; yes, W(ef, e—f, e2f, e—2f) — 0, —ζ < t < ζ
W — ce-2f W — c/f2
27. / = c1ef + c2t + c3fef Section 4.2, page 219
1 V2 el[(n/4)+2mn ]
3 3^+2mn)
5 2ei[(11n/6)+2mn ]
7. 1, 2 (—1 + iͺ3), 2 (—1 — iͺ3)
9. 1 , i2, — 1 , — i 2
11. y — c1ef + c2tef + c3e—f
13. y = c1ef + c2e2f + c3e— f
15
16
17
18
19
20 21
22
22. W = c 24. W — c/t
28. y — c1 f2 + c213 + c3(t + 1)
2 2ei [(2n/3)+2mn]
4 ei [(3n/2)+2mn]
6 V2ei [(5n/4)+2mn]
8. 21/4e—"³/8, 21/4e7ni/8
10. (V3 + i)/V2, — (V3 + J)/V2
12. y — c1ef + c2tef + c3t2ef 14. y — c1 + c2t + c3e2f + c4fe2f
t + c2 sin t + Ί^/2 (c3 cos 2 t + c4 sin 11) + e v^f/2 (c5 cos 11 + c6 sin 11)
y — c1 cos f i ^2 sin l i e (°3 cos 2 f i °4 sui 2 f
y — c1ef + c2e— f + c3e2f + c4e—2f y — c1ef + c2tef + c312 ef + c4e—f + c5te— f + c6t2e—f
y — c1 + c2t + c3ef + c4e—f + c5 cos t + c6 sin t
t 2t
y — c1 + c2e + c3e + c4 cos t + c5 sin t y — c1 + c2e2f + e—f (c3 cos \/3 t + c4 sin \/3 t)
y — ef[(c1 + c21) cos t + (c3 + c4t) sin t] + e— f[(c5 + c6t) cos t + (c7 + c8t) sin t]
y — (c1 + c2t) cos t + (c3 + c4f) sin t 23. y — c1 ef + c2e'
24. y — c1e—f + c2e(—2+'^2)f + c3e(—2—'^)f
25. y — c1e—f/2 + c2e— f/3cos(f/ͺ) + c3e—f/3 sin(t/ͺ)
26. y — cje3f + c2e—2f + c3e(3+e3)f + c4e(3—e3)f
,(2+V5)f + (2—V5)f
u
6
e
694
Answers to Problems
See SSM for detailed solutions to 27 ,29 ,30, 31
34,37,38a, 39ac
1
5, 9, 13
17, 20, 22a
22be
1,4, 5, 7
27. σ = c1e f/3 + c2e f/4 + c3e f cos2f + c4e f sin2f
28. σ = c1 e— f cos f + c2e— f sin f + c3e—2f cos(\/3 f) + c4e— 2f sin^\/3 f)
29. σ = 2 —2cosf + sin f
30. σ = 2 e— f/v^ sin(f/^2) — 2ef/'^2 sin(f/V2)
31. σ = 2f— 3 32. σ = 2cosf— sin f
33. Σ = — 3ef— ^e2f— 6e—2f— ^e— f/2 34. σ = ^e— f + 23ef/2cos f + -3
35. σ = 8 —18e—f/3 + 8e—f/2
36. σ = 23 e—f cos f — 38 e— f sin f — 13e—2f cos(V3 f) +-----e—2f sin(V3 ?)
37. σ = 2 (cosh f — cos f) + 2 (sinh f — sin f)
38. (a) W = c, a constant (b) W = — 8 (c) W = 4
39. (b) u1 = c1 cos f + c2 sin f + c3 cosV6 f + c4 sin\/α f
Section 4.3, page 224
1. σ = c1ef + c2fef + c3e— ‘ + 1 fe— f + 3
2. σ = c, ef + c2e— f + c3 cos f + c4 sin f — 3 f — 4 f sin f
3. σ = c,e—f + c2 cos f + c3 sin f + 2 fe— f + 4(f — 1)
4. σ = c1 + c2ef + c3e— f + cos f
5. Σ = c1 + c2f + c3e— 2f + c4e2f— 1 ef— 48f4 — ^f2
6. σ = c1 cos f + c2 sin f + c3f cos f + c4f sin f + 3 + 9 cos2f
7. σ = c1 + c2f + c3f2 + c4e—f + ef/2[c5 cos^V3 f/2) + c6 sin^V3 f/2)] + ^ f4
8. σ = c1 + c2f + c3f2 + c4e— f + 20 sin2f + 40 cos2f
9. σ = 16(1—cos2f) + 1 f2
10. σ = (f — 4) cos f— (2 f + 4) sin f + 3f + 4
11. σ = 1 + 1 (f2 + 3f) — fef
12. σ = — 5 cos f — 5 sin f + 20e—f ^ef + 53e—3f + 6Z cos2f — I9 sin2f
13. Σ (f) = f (A0f3 + A,f2 + A2f + A3) + Bf2ef
14. Σ(f) = f (A0f + A,)e— f + 5cos f + Ρ sin f
15. Σ(f) = Af2e‘ + Bcos f + Ρ sin f
16. Σ (f) = Af2 + (B0f + B,)ef + f (C cos2f + D sin2f)
17. Σ(f) = f(A0f2 + A,f + A2) + (B0f + Bf) cos f + (C0f + Ρ,) sin f
18. Σ(f) = Aef + (B0f + B,)e— f + fe — f(Ccos f + Dsin f)
19. *0 = ^0’ kn = ^0an + a,an 1 +••• + an— ,a + an
Section 4.4, page 229
1. σ = c, + c2 cos f + c3 sin f — ln cos f — (sin ?) ln(sec f + tan f)
2. σ = c, + c2ef + c3e— f — 2f2
3. σ = c,ef + c2e— f + c3e2f + 30e4f
4. σ = c, + c2 cos f + c3 sin f + ln(sec f + tan f) — f cos f + (sin f) ln cos f
5. σ = c, ef + c2 cos f + c3 sin f — f e— f cos f
6. σ = c, cos f + c2 sin f + c3 f cos f + c4 f sin f — 1 f2 sin f
7. σ = c, ef + c2 cos f + c3 sin f — 2 (cos f) ln cos f + f (sin f) ln cos f — 2 f cos f —
+ 2 e\ft (e— s/ cos s^ ds
ei/2 sin f
2 f sin f
Answers to Problems
695
See SSM for detailed solutions to 11, 14
16
8. σ = c1 + c2el + c3e * — lnsin t + ln(cos t + 1) + 2el ? ^e s/ sins^
+ 2 e-t^" / sin s^ ds
9. c1 = 0, c2 = 2, c3 = 1 in answer to Problem 4
10. c1 = 2, c2 = 7, c3 = — 7, c4 = 1 in answer to Problem 6
11. c1 = 2, c2 = 2, c3 = —|, t0 = 0 in answer to Problem 7
12. c1 = 3, c2 = 0, c3 = —Ίλ/2, t0 = ο/2 in answer to Problem 8
13. Y (x) = x4/15
14. Y(f) = 1 f [et-s — sin(t — s) — cos(t — s)]g(s) ds
2 Jt0
15. Y(t) = 1 f [sinh(t — s) — sin(t — s)]g(s) ds
Jt0
16. Y(t) = 1 f e(t-s')(t — s)2g(s) ds; Y(t) = —tel ln |t|
2 Jt0
17. Y(x) = 2 f x[(x/t2) - 2(x2/t3) + (x3/t4)]g(t) dt
2 J x,.
ds
CHAPTER 5 Section 5.1, page 237
2, 5, 9, 12, 13
1. p = 1
3. p — ζ
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