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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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y1(x) = ^ ^ + 3x + -r + ... + n!(-^ + .
10. r2 - r + 4 = 0; (n + r - 2)2an + an-2 = 0; r1 = r2 = 1/2
x2 x4 (-1)mx2 m
----I---------------L -------------------------------
22 2242 22m (m!)2
11. r2 = 0; r1 = 0, r2 = 0
y1 (x) = x1/^ 1 - -2 + -2-2--------------+ 02m, , \2 + •
, , a(a + 1) f a(a + 1)[1 • 2 - a(a + 1)], ^2, σ (x) — 1 I---------2— (x — 1)-------------------2-----2-(x — 1) + •••
1 2 • 12 (2 • 12)(2 • 22)
,/ nn+1 a(a + 1)[1 • 2 - a(a + 1)] ••• [n(n - 1) - a(a + 1)]
+ <"1) ------------------------------------------------------------------(x -1) +'
12. (a) r1 = 2, r2 = 0 at both x =±1
(b) y1(x) = |x - 1|1/2
Ζ (—1)n(1 + 2a) •••(2n — 1 + 2a)(1 — 2a) •••(2n — 1 — 2a)
1 + g----------------------------------------------------------------------(x - 1)
-ζ (-1)n a(1 + a) •••(n - 1 + a)(-a)(1 - a) • • • (n - 1 - a)
72 ) = 1 + g---------------------------n!1 35-• • (2n - 1)--------------------------(x - 1)
13. r2 = 0; r1 = 0. r2 = 0; a = ——1 2k)an—1
1 2 n n2
—κ (—Λ)(1 — X) 2 (—Λ)(1 — X) • • • (n — 1 — X) n
y1 (x) = 1 I--------2 x I--------2----x2 I----------------------------------------------1-2-x I-
1 (1!)2 (2!)2 (n!)2
For κ = n, the coefficients of all terms past xn are zero.
16. (b) [(n — 1)2 — 1]bn = —b 2, and it is impossible to determine b2.
702
Answers to Problems
See SSM for detailed solutions to 1, 3, 9
17abc
18
20abc, 21 bd 1 2
Section 5.7, page 278
1. x--- 0; r (r --- 1) --- 0; r1 --- 1, r2 --- 0
2. x--- 0; r2 --- 3r + 2 --- 0; r1 --- 2, r2 --- 1
3. x--- 0; r (r --- 1) --- 0; r1 --- 1, r2 --- 0
x--- 1; r (r + 5) --- 0; r1 --- 0, 5
4. None


5. x--- 0; r2 + 2r --- 2 --- 0; r1 --- -1 + V3 --- 0.―32, r2
6. x--- 0; (r r1 3 2r
r --- 4 ---
1 0
443
---
;0
x--- -2 ;--- 0; r1 --- 0
4--- ---
r( ---2 r2
r(
―. x--- 0; r2 + 1 --- 0; r1 --- i, r2 --- -³
8. x--- -1 ; r2 --- 7r + 3 --- 0; r1 --- (― + V37)/2 --- 6.54
9. x--- 1; r2 + r --- 0; r1 --- 0, r 2------1
10. x--- -2 ; r2 --- (5/4)r --- 0; r1 --- 5/4, r2 --- 0
11. x--- 2; r2 --- 2r --- 0; r1 --- 2, r2 --- 0 2
x--- -2 r2 ;r 1---2 0
--- II
2 ---2 r
II
0
12. x--- 0; r2 --- (5/3)r --- 0; r1 --- 5/3, r2 --- 0
x — —3; r2 - (r/3) - 1 — 0; r1 — (1 + *Δ―)/6 — 1.18, r2 — (1 — -Δ―)/6 = —0.84― 13. (b) r1 — 0, r2 — 0
(c) Σ1(υ) — 1 + X + 4 x2 + 36 x3 + •••
Yi(x) — Σ1(υ) x — 2x — 4x2 — 18x3 + .
14. (b) r1 — 1 , r2 — 0 (c) y1(x) — x — 4x2 + 17 x3 — 12 x4 + •••
y2(x) — -6y1 (x) lnx + 1 — 33x2 + 4Px3 + •••
15. (b) rj — 1, r2 — 0
(c) y1(x) — x + 2x2 +9 x3 +51 x4 + •••
y2(x) — 3y1(x) lnx + 1 — f x2 — 19x3 + •••
16. (b) r1 — 1, r2 — 0
(c) y1(x) — xr — 2 x2 + 112 x3 — x4 + •••
y2(x) — —y^x) lnx + 1 — 4x2 + 36x3 — ^x4 + •••
1―. (b) r1 — 1, r2 — -1
(c) y1(x) — x — 24x3 + ―ιx5 + •••
y2(x) — — 3y1 (x) lnx + x—1 — 90x3 +--------------
18. r1 — 5 - r2 — 0
y1(x) — (x — 1)1/2[1 — 4 (x — 1) + 480 (x — 1)2 + •••], p — 1
19. (c) Hint: (n — 1)(n — 2) + (1 + a + β)(ο — 1) + ΰβ — (n — 1 + a)(n — 1 + β)
(d) Hint: (n — Y)(n — 1 — y) + (1 + a + e)(n — σ) + ΰβ — (n — σ + a)(n — σ + β)
Section 5.8, page 289
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