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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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a2 — ~(a0 + a1)/2
y1(x) — 1 — 1 (x — 1)2 + 1 (x — 1)3 — 112 (x — 1)4 + •••
Ó2 (x) — (x — 1) — 2 (x — 1)2 + 6 (x — 1)3 — 6 (x — 1)4 + •••
9. (n + 2)(n + 1)an+2 + (n — 2)(n — 3)an — 0; n — 0, 1, 2,...
y1 (x) — 1 — 3x2, y2(x) — x — x3/3
10. 4(n + 2)an+2 — (n — 2)an — 0; n — 0, 1, 2,...
x2
x3 x5
y1(x 1 4 , y2(x) x 12 240 2240
x2 n|1
4n (2n — 1)(2n + 1)
7
a3 — 4 a1
Answers to Problems
697
See SSM for detailed solutions to 14, 16a, 19
22b, 23, 26
1,6, 9afh
11. 3(n + 2)an+2 — (n + 1)an = 0; n = 0, 1, 2,...
yf(x) = 1 + x! + x! + _l x6 + •••+ 3„ 5"-(2n~f) x2 n+•..
ë 6 24 432 3n • 2 • 4 ••• (2n)
2 3 8 5 16 7 2 • 4 • • • (2n) ,
ê, (x) = x + - x +--------------------------x5 +----------------x7 +--------------------------------+ —-x 1 + •
Ó³ 9 135 945 3n • 3 • 5 ••• (2n + 1)
12. (n + 2)(n + 1)an+2 — (n + 1)nan+, + (n — 1)an = 0; n = 0, 1, 2,...
x2 x3 x4 xn y-(x) = 1 + Ó + Ó + 24 + ••• + d + •••, y2= x
13. 2(n + 2)(n + 1)an+2 + (n + 3)an = 0; n = 0, 1, 2,...
y,(x) = 1 — 3 x2 + — x4 — — x6 + ••• + ( — 1)n 3 ^ 5 "n(2n + -) x2 n + • •
ë 4 32 384 2n (2n)!
y,(x) = x — x3 + x5 — .x! + ••• + (—,)n (2n + 2) x2 n+- + •••
Ó³ 3 20 210 2n (2n + 1)!
14. 2(n + 2)(n + 1)an+2 + 3(n + 1)an+, + (n + 3)an = 0; n = 0, 1, 2,...
y,(x) = 1 — I(x — 2)2 + 3 (x — 2)3 + 64(x — 2)4 + • • •
y2(x) = (x — 2) — 3 (x — 2)2 + 24 (x — 2)3 + 64 (x — 2)4 + • • •
15. (a) ó = 2 + x + x2 + -x3 + -x4 + • • • (c) about |x| < 0.7
16. (a) ó =— 1 + 3x + x2 — 3x3 — 6x4 + • • • (c) about |x| < 0.7
17. (a) ó = 4 — x — 4x2 + 2x3 + 3x4 + • • • (c) about |x| < 0.5
18. (a) ó = —3 + 2x — 3x2 — 2x3 — 8x4 + • • • (c) about |x| < 0.9
19. y,(x) = 1 — f(x — 1)3 — -2(x — 1)4 + -8(x — 1)6 + • • •
y2(x) = (x — 1) — 1 (x — 1)4 — 20 (x — 1)5 + 28 (x — 1)7 + • • •
T! / Ë , 1 k 2 , k(k — 4) 4 k(k — 4)(k — 8) 6 ,
21. (a) ó (x) = 1----------x2 H-----------------------------------------x4-x6 + • • •
1 2! 4! 6!
ê — 2 3 (k — 2)(k — 6) 5 (ê — 2)(k — 6)(k — 10) 7 y2(x) = x — x +-5-x5---------------------------------------------------------x + •
(b) 1, x, 1 — 2x2, x — 3x3, 1 — 4x2 + 3x4, x — 4x3 + -5x5
(c) 1, 2x, 4x2 — 2, 8x3 — 12x, 16x4 — 48x2 + 12, 32x5 — 160x3 + 120x
22. (b) ó = x — x3/6 +• • •
Section 5.3, page 253
1. ô"(0) = -1,
2. ô"(0) = 0,
3. ô"(1) = 0,
4. ô"(0) = 0,
Ô "'(0) = 0, ô '"(0) = -2, Ô '"(1) = -6, ô '"(0) = -a0,
ôlv (0) = 3 ôlv (0) = 0 ôlv (1) = 42 ô-(0) = —4a,
5. ð = æ, ð = æ
6. ð = 1, ð = 3, ð = 1
7. ð = 1, ð = -/!
8. ð = 1
9. (a) ð = æ (b) ð = æ (c) ð = æ
(f) ð = V2 (g) ð = æ (h) ð = 1
(k) ð = V3 (l) ð = 1
(d) ð = æ (e) ð = 1
(³) ð = 1 (j) ð = 2
(m) ð = æ (n) ð = æ
698
Answers to Problems
See SSM for detailed solutions to 10a
10b, 11
18
20, 22
26, 28
a2 2 (22 - a )a 4 (42 - a2)(22 - a2)a
l0- (a)7.M = 1 - -x2 - * „ ' x4 - '-^->-
Ó
-x2 m-----
[(2m - 2)2 - a2] ••• (22 - a2)a2
(2m)!
1 — a2 3 (32 — a2)(1 — a2) 5
72(x) = x + ~3à x3 + ^^--------------------->- x5 + •••
, [(2m - 1)2 - a2] •••(1 - a2) x2m+1 ,
(2m + 1)!
(b) y1 (x) or y2(x) terminates with xn as a = n is even or odd.
(c) n = 0, ó = 1; n = 1, ó = x; n = 2, ó = 1 — 2x2; n = 3, ó = x — 3x3
Ï. 71(x) = 1 - 6x3 + ^x5 + 4x6 + ••• , 72(x) = x - 112x4 + æx6 + s0i4x7 + ¦
p = æ
12. 71(x) = 1 - 6x3 + 12x4 - 45x5 + •••. 72(x) = x - 12x4 + 20x5 - 6¯¯x6 + ••• >
13. 71(x) = 1 + x2 + ³x4 + ^x6 + •••, y2(x) = x + 6x3 + gLx5 + ^x7 + •••, p = n/2
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