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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
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y1(x) = ^ y + 3x + T + • • • + f(-+^ +
10. r2 - r + 4 = 0; (n + r - 2)2an + an-2 = 0; r1 = r2 = 1/2
x2 x4 (-1)mx2 m
1 • • • + -------------------
22 2242 22m (m!)2
11. r2 = 0; r1 = 0, r2 = 0
y1 (x) = x1/^ 1 - -2 + 7^2-----------------+ 02m, ,\2 + •
, , a(a + 1) r ^ a(a + 1)[1 • 2 - a(a + 1)^ ^2|
y, (x) — 1--------------------------+--------------------2— (x — 1 )-2-2-(x — 1 ) + • • •
1 2^12 (2 • 12)(2 • 22)
,/ tna+1 a(“ + 1)[1 • 2 — a(a + 1)]-• • [n(n — 1) — a(a + 1)1
+ <"l) -------------------------------?w-------------------------------(x —1) +'
12. (a) r1 — 2, r2 — 0 at both x —±1
(b) y1(x) — |x — 1|1/2
(—1)n(1 + 2a) • • • (2n — 1 + 2a)(1 — 2a) • • • (2n — 1 — 2a)
1 + g---------------------------------Fc??+n-------------------------------(x —
(—1)na(1 + «)• • • (n — 1 + a)(—a)(1 — a) • • • (n — 1 —
y2(x) — 1 + n—1-----------------------------------------------------n!1 .3 .5. • • (2n — 1)----------------------------------------------(x — 1)
13. v- — 0; r1 — 0. v2 — 0; a — L—1 2
1 2 n n2
—X (—X)(1 — X) 2 (—X)(1 — X) • • • (n — 1 — X) n
y1 (x) — 1 +------2 x +------------------2----------------x2---------+--------1----2---x---+--
1 (1!)2 (2!)2 (n!)2
For X — 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, 21bd 1
2
Section 5.7, page 278
1. x= 0; r (r - 1) = 0; r1 = 1, O II C
2. x= 0; r2 - 3r + 2 = 0; r1 = 2, r2 = 1
3. x= 0; 1 II o r1 = 1, O ' Il
x= 1; r (r + 5) = 0; r1 = 0, ln 1 II
4. None
5. x= 0; r2 + 2r - 2 = 0; r1 = -1 + V3 = 0.732, r2
6. x= 0; 1 -Mu> II O r1 3 = 4 o II
x= -2 ; r (r - 5 ) = 0; r1 = o II
7. x= 0; 10 + 11 O r1 = i, r2 = —i
8. x= -1 ; r2 - 7r + 3 = 0; r1 = (7 + V37)/2 = 6.54
9. x= 1; r2 + r = 0; r1 = 0, r = -1
10. x= -2 ; r2 - (5/4)r = 0; r1 o II in II
11. x= 2; r2 - 2r = 0; r1 = 2, o II
x= -2 ; r2 - 2r = 0 ;r = 2 o II
12. x= 0; r2 - (5/3)r = 0; r1 = 5/3- r2 = 0
x =-3; r2 - (r/3) - 1 = 0; r1 = (1 + v/37)/6 = 1.18,
r2 = (1 - V37)/6 = -0.847 13. (b) r1 = 0, r2 = 0
(c) y?x) = 1 + X + 4x1 + 36x3 + •••
Yi(x) = y1(x) x - 2x - 4x2 - x3 + •
14. (b) r1 = 1, r2 = 0
(c) y1(x) = x - 4x2 + 137 x3 - 12 x4 + •••
y2(x) = -6y1 (x) lnx + 1 - 33x2 + 4Px3 + •••
15. (b) = 1, r2 = 0
(c) y1(x) = x + 2 X2 + 9 X3 + 51 x4 + •••
y2(x) = 3y1(x) lnx + 1 - f x2 - 19x3 + •••
16. (b) r1 = 1, r2 = 0
(c) y1(x) = x - 2 x2 + 112 x3 - pi x4 + •••
y2(x) = -y (x) ln x + 1 - 4x2 + 36x3 - pig x4 + •••
17. (b) r1 = 1, r2 = -1
(c) y1(x) = x - 24x3 + tTo x5 + •••
y2(x) = 3y1 (x) lnx + x-1 - 90x3 +-------------
18. r1 = 5 - r2 = 0
y1(x) = (x - 1)1/2[1 - 4 (x - 1) + 4m (x - 1)2 + •••]- p = 1
19. (c) Hint: (n - 1)(n - 2) + (1 + a + ?)(n - 1) + a? = (n - 1 + a)(n - 1 + ?)
(d) Hint: (n - Y)(n - 1 - y) + (1 + a + ?)(n - y) + a? = (n - y + a)(n - y + ?)
Section 5.8, page 289
(-1)nxn
1 y1(x) S n!(n + 1)!
1
y2(x) = -y1(x) ln x +
~ h + H
1 - y _2---------n-i (-1)nx"
n! (n - 1)!
x
1 - (-1)nxn 2 ? (-1)n_
y1(x) = “E -^^2- - y2(x = y1(x) ln x - “E '
n=0
(n!)2 ^2 ^ x^T (n!)2
xn
2
Answers to Problems
703
See SSM for detailed solutions
to 3, 4, 5
7, 12
13, 14
to ( i \n^n to (_1 )n2n h
3 y1(x) = E - n2 xn, y2(x) = y1(x)lnx-2E ^HLxn
1 to
4. y1(x) = -E
n=0
n=0 (n!)2
, _(-1)n
x
n=1
(n!)2
n! (n + 1 )!
xn
1
y2(x) = -y1 (x) ln x + xi
to h + h
1 -? Hn(+ Hn-1 (-1)nxn
5. y1(x) = x3/2
1+
n=1 n! (n - 1)! (-1)"
x \2m
1 m!(1 + 2)(2 + §) ? ? ? (m + 2)
(!)
y2(x) = x
x-3/2
1+
(-1)m
=1 m ! (1 - 2 )(2 - 2 )? ? ? (m - 3 ) V 2
*( D2
Hint: Let n = 2m in the recurrence relation, m = 1, 2, 3,.. is arbitrary.
. For r = — 2, a1 =
CHAPTER 6 Section 6.1, page 298
1, 2, 5b, 6, 9
13, 16, 19, 21
25,27abcd
2, 4, 7
11, 14, 15
1. Piecewise continuous 4. Piecewise continuous
2. Neither
5. (a) 1 /s2, s > 0
(b) 2/s3, s > 0
(c) n! /s
3. Continuous
6. s/(s2 + a2),
?n+1
7.
9.
11.
13.
15.
17.
19.
22 s — b
s > | b|
2 -2, s - a > |b|
s > 0 b
(s - a)2 - b b
2, s > 0
s2 + b2
b
(s - a)2 + b2 ,
1
(s - a)2
22 s + a
(s - a)2(s + a)2’
s > |a|
2a (3 s2 - a2) (s2 + a2)3 21. Converges 23. Diverges
s > 0
10.
12.
14.
16.
18.
20.
22 s - b
b
(s - a)2 - b s
s > |b|
2 -5, s - a > |b|
s2 + b2
s > 0
(s - a)2 + b2 ’
2as
(s2 + a2)2
n!
(s - a)n+1, 2a(3s2 + a2) (s2 - a2)3 :
s > 0
s > |a|
22. Converges 24. Converges
26. (d) r (3/2) = 4n/2; Y (11 /2) = 945jn/32
Section 6.2, page 307
1. 2sin2t
3. §ft - §e-4t 5. 2e-t cos2t 7. 2et cos t + 3et sin t 9. -2e-2t cos t + 5e-2t sin t 11. y = 1 (e3t + 4e-2t)
13. y = et sin t
2. 2tV
4. 9 e3' + 5 e-2t 6. 2cosh2t - |sinh2f 8. 3 - 2sin2t +5cos2t 10. 2e-t cos3t - 3 e-t sin3t 12. y = 2e-t - e-2t 14. y = e2t - te2
15. y = 2et cosh^/? f - (2^\f?)et sinh\/3 t 16. y = 2e ^ cos2t + 2 e ^ sin2t
s
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
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