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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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(—1)nxn
1 y1(x) n—0 n!(n + 1)!
1
y2(x) — —y1 (x) ln x +
H + H
1 — σ _n-n—³ (—1)nxn
n—1 n! (n — 1)!
x
1 - (—1)nxn 2 ^ (—1)n_n
y1(x) — “E -^^2- - y2(x — y1(x) ln x — 7E '
(n!)2 "2 ^ x^1 (n!)2
xn
2
Answers to Problems
703
See SSM for detailed solutions to 3, 4, 5
7, 12
ζ (___³ \n^n ζ (__- )n2n h
3. yi(x) = J2 — n2 xn, Σ2(x) = yi(x)lnx — 2J2--------------------------^HLxn

4. y,(x) = -E
n=0 (n!)2
ζ (—1)n
n=1
(n!)2
x n=0 n! (n + 1 )!
1
y2(x) = -y,(x) lnx + x2
xn
ζ h + H 1 — ? Hn+ H—f (—1^
5. y,(x) = x3/2
1+
n=f n!(n — 1)! (—1)m
x \2m
f m! (1 + 2)(2 + 2) • • • (m + 3)
(2)
Σ2(Υ) = X
x—3/2
1+
(—1)m
=f m! (1 — 2 )(2 — 2 )• • • (m — 3 ) V 2
*( D2
13, 14
Hint: Let n = 2m in the recurrence relation, m = 1, 2, 3,.... For r is arbitrary.
= 2, ai _
CHAPTER 6 Section 6.1, page 298
1,2, 5b, 6, 9
13, 16, 19, 21
25,27abcd
1. Piecewise continuous 4. Piecewise continuous
2. Neither
5. (a) 1 /s2, s > 0
(b) 2/s3, s > 0
(c) n! /s
7.
9.
11.
13.
15.
17.
19.
22 s — b
s > | b|
(s — a)2 — b2 ’ b
s > 0
s — a > |b|
s2 + b2 b
(s — a)2 + b2 ’
1
(s — a)2
22 s + a
(s — a)2(s + a)2’ 2a(3s2 — a2)
(s2 + a2)3 ,
s > |a|
s > 0
21. Converges 23. Diverges
3. Continuous
6. s/(s2 + a2),
¦n+l
s > 0 b
22
sb
s > | b|
10.
12.
14.
16.
18.
20.
b
(s — a)2 — b2 s
s — a > |b|
s2 + b2
s > 0
(s — a)2 + b2 ’
2as
(s2 + a2)2
n!
(s — a)n+f, 2a(3s2 + a2) (s2 — a2)3 :
s > 0
s > |a|
22. Converges
24. Converges
26. (d) Γ (3/2) = 4Λ/2; Γ (11 /2) = 945σΟ/32 Section 6.2, page 307
2, 4, 7
11, 14, 15
1. 3sin2t
3. § ft — 5 e-4t 5. 2e—t cos2t 7. 2et cost + 3et sint 9. —2e—2t cos t + 5e—2t sin t 11. σ = 1 (f3t + 4e—2t)
13. σ = e( sin t
2. 2tV
4. 9 e3t + 5 e—2t 6. 2cosh2t — |sinh2f 8. 3 — 2sin2t +5cos2t
10. 2e c cos3t — 3 e 1 sin3t
12. y = 2f—t — f—2t
14. σ = e2t — te2
15. σ = 2el cos^V! f — (2/\[Ώ)ε³ sin^v',3 t 16. σ = 2e ΄ cos2t + 2 e ΄ sin2t
s
sa
sa
s > a
s > a
s > a
s > a
0 and a3
s > 0
704
Answers to Problems
See SSM for detailed solutions to 17, 20, 22, 24
27b, 30, 32
36a,38ab 2, 4, 8
14, 21,22,27
28, 30
1
3
1―. y — tet — t2e* + 2t3e* 18. y — cosh t
19. y — cosV2 t
20. y — (a2 — 4)—1 [(rn2 — 5) cos at + cos 2f]
21. y — 1 (cos t — 2 sin t + 4Ί* cos t — 2Ί* sin t)
22. y — 5 (e—'t — Ί* cos t + 7Ί* sin t)
s 1 — e—n 5
24. Σ (s) — -1----2-------
s2 + 4 s(s2 + 4)
26. Σ (s) — (1 — e—5 )/s2(s2 + 4) 29. 1 /(s — a)2
30. 2b(3s2 — b2)/(s2 + b2)3 31. n!/5
23. y — 2e—* + te-* + 2t2e—t
1 e—s(s + 1)
25. Σ (s) — —2—2------------2—2----------
s2(s2 + 1) s2(s2 + 1)
n+1
32. n!/(s — a)
n+1
33. 2b(s — a)/[(s — a)2 + b2]2
34. [(s — a)2 — b2]/[(s — a)2 + b2]2
36. (a) Σ + s2y — s (b) s2Σ" + 2sy' — [s2 + ΰ(ΰ + 1)]Σ —-1
Section 6.3, page 314
7. F (s) — 2e—s/s3 8.
e—Λ s e—2ns
9. F(s) — —2---------------(1 + n s) 10.
F(s) — e—s(s2 + 2)/s3
1
2 — 2 s2 s2
11. F (s) — s—2[(1 — s)e—2s — (1 + s)e—3s 13. f(t) — ?3e2t 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. f (t) — 1 et/3(e2t/3 — 1)
24. F(s) — s—1(1 — e—s), s > 0
25. F (s) — s—1 (1 — e—s-
26. F(s) — 1[1 — e—s +
1 Ζ
27. F(s) — - ?(—1)ne—ns —
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