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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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14. 71(x) = 1 + 1 x3 + 12x4 - ๘x6 + •••, 72(x) = x - 1 x3 + 24x4 + 120^ + ••• -
p=1
15. Cannot specify arbitrary initial conditions at x = 0; hence x = 0 is a singular point.
x2 xn
16. ๓ = 1 + x + — + ••• + — + ••• = ex
2! n!
x2 x4 x6 x2n
17. 7 = 1 + T + ร4 + ฯาธ + ••• + + •••
18. ๓ = 1 + x + 2x2 + 2x3 I------
19. ๓ = 1 + x + x2 + •••+ xn + •••=-----------
1 - x
( x2 xn — (x3 x4 xn —
20. 7 — ai 1 + x + ——I ••• I------------- + •••) + 2 I ——I—— + ••• I------ + ••• I
y 0\ 2! n! / \3! 4! n! I
= ae + 2( ex - 1 - x - ๓ ) = ce" - 2 - 2x - x2
x2 x4 x6 (-1)nx2n
21. 7 — a0 I 1 —ใ—I—2------------------------3---1 ••• I-~n—:------------------------------1 ฆ
0 \ 2 222! 233! 2nn!
x2 x3 x4 x5 -
+ ( x + — — — —--------------1-------1 •
2 3 2 • 4 3 • 5
2/2 ( x2 x3 x4 x5
= a0 e-x /2 + x I-----------------------1-------1 •
0 ๓ 2 3 2 • 4 3 • 5
23. 1,1 - 3x2, 1 - Dx2 + fx4; x, x - 5x3, x - 14x3 + 21 x5
24. (a) 1, x, (3x2 - 1)/2, (5x3 - 3x)/2, (35x4 - 30x2 + 3)/8, (63x5 - 70x3 + 15x)/8
(c) Pp 0; P2, ฑ0.57735; P3, 0, ฑ0.77460; P4, ฑ0.33998, ฑ0.86114;
P5, 0, ฑ0.53847, ฑ0.90618
Answers to Problems
699
Section 5.4, page 259
17, 21, 21เ, 22, 25, 31
See SSM for 1. x --- 0, regular 2. x --- 0, regular; x --- 1, irregular
detailed solutions 3. x --- 0, irregular; x --- 1, regular 4. x --- 0, irregular; x ---ฑ1, regular
to 1, 5 5. x --- 1, regular; x --- --- 1, irregular 6. x --- 0, regular
7. x --- --- 3, regular 8. x --- 0, 1, regular; x --- 1, irregular
9. x --- 1, regular; x --- --- 2, irregular 10. x --- 0, 3, regular
12, 17 11. x --- 1, --- 2, regular 12. x --- 0, regular
13. x --- 0, irregular 14. x --- 0, regular
15. x --- 0, regular 16. x --- 0, ฑnn, regular
19, 21, 23 17. x --- 0, ฑnn, regular 18. x --- 0, irregular; x --- ฑnn, regular
19. ( x2 x4 \
y--- a^1 2 • 5 + 2 • 4 • 5 • 9 • • )
22. Irregular singular point 23. Regular singular point
25 24. Regular singular point 25. Irregular singular point
26. Irregular singular point 27. Irregular singular point
Section 5.5, page 265
2, 4, 9, 13, 16 1. 12 2. y --- c1|x + 1|--- 1/2 + c2|x + 1|--- 3/2
3. y --- c1 x + c2 x | x| )
4. y --- c1 x2 + c2x2 ln |x|
y --- c1 x---1 cos(2 ln |x|) + c2x--- 1 sin(2 ln
5. y --- c1 x + c2x ln |x| 6. y --- c1(x--- 1)---3 + c2(x--- 1)---4
7. y --- dlx |(--- 5+^)/2 + c2|x |(---5---^~9)/2
9. y


2x
10. y—
11. y—
12. y— 14. y — 16. y — 18. โ >
20. เ >
21. (a)
(b)
(c)
(d)
(e)
24. y — 26. y — c1
|x|3/2 cos( 1\/31๏ |x|) + c2|x|3/2 sin(2v/3ln |x|) x3 + c2x3 ln |x|
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