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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.
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max|e| = 1.6025 at x = ±n max|e| = 1.5867 at x = ±n max|e| = 1.5788 at x = ±n
(c) Not possible
8. (a)
(b)
1 ë OO ³
1 2 1 — cos nn
f (x) = - +------------T Ó ----------------~------------cos nn x
ï 2 n2
2
(c) 9. (a)
(b)
(c)
n = 10 n = 20 n = 40 n = 21
max|e| = 0.02020 at x = 0, ±1 max|e| = 0.01012 at x = 0, ±1 max|e| = 0.005065 at x = 0, ±1
2 ^ (-1)n+1 f (x) = — > --------------sin nn x
Ò¥ ' Ï
n=1
n = 10, 20, 40; max|e| = 1 at x ± 1
Not possible
33
n ï
2
44 n ï
22 n ï
n even
a
0
n
726
See SSM for detailed solutions to 12a
12b, 14
15, 16, 18a 18b
3, 6
7, 10, 13, 14 15, 18
1 Æ
10. (a) f(x) = 2 +
2 n=1
6(1 — cos nn) nnx 2cos nn . nnx
22 n n
¦ cos--------1-
2
¦ sin -nn 2
(b) n = 10 n = 20 n 40
lub|e| = 1.0606 as x ^ 2 lub|e| = 1.0304 as x ^ 2 lub|e| = 1.0152 as x ^ 2
(c) Not possible
1 Æ
11. (a) f(x) = - + J2
2 cos nn
2 — 2 cos nn + n2n2 cos nn
¦ cos nnx —
(b) n = 10 n = 20 n 40
lub|e| = 0.5193 as x ^ 1 lub|e| = 0.5099 as x ^ 1 lub|e| = 0.5050 as x ^ 1 (c) Not possible
12 ft \ 12 (—1)n ¦
12. (a) ³ (x) =---------3 > —3— sin nn x
n n=1 n
(b) n = 10 n = 20 n = 40
(c) n = 4
max|e| = 0.0013 45 at x = ±0.973 5 max|e| = 0.0003534 at x = ±0.9864 max|e| = 0.00009058 at x = ±0.9931
13. ó = (a sin nt — n sin at)/a(a2 — n2), a2 = n2 ó = (sin nt — nt cos nt)/2n2, a>2 = n2
Æ
14. ó = bn (a sin nt — n sin at)/a(a2 — n2), a = 1, 2, 3,...
n=1
æ
7= E bn(m sin nt — n sin mt)/m(m2 — n2) + bm (sin mt — mt cos mt)/2m2, a
n= 1
n=m
15. Ó =
Ï a2 — (2n — 1)2
11
---sin(2n — 1)t-sin at
2n 1 a
16. 7 = cos a
cos(2n — 1)n t — cos at n2 (2n — 1)2[a2 — (2n — 1)2]
Section 10.4, page 570
1. Odd
4. Even
2. Neither 5. Even
3. Odd 6. Neither
14 -Æ 1 - cos(nn/2) nnx
14. f (x) = 7 + —^----------------------------2-cos-
4
n=1
2
4 (nn/2) — sin(nn/2) nn x
f (x) = "1 > -----------------2----------sin -
n 2 n=1 n2
2
1 2 -Æ
15 f(x) = 2 + -L
(-1)n
(2n — 1)n x
2 ^ , 2n — 1
n= 1
2 / 2 n^ \ nn x
16. f (x) = ó — — cos nn +------------------------sin ) sin -
< J ÏÒ¥ \ 11 TJ "
n=1
17. f (x) = 1
¦ x—v
18. f (x) = -Y]
77" -4
4 ^ sin(2n — 1)x
n=1
2n 1
n=1
2 / nn 2nn
19. f (x) = ^ — ( cos-----------------+ cos-----------2
cos nn sin
22 n Ï
33
n Ï
1
cos
nx
3
=m
727
See SSM for detailed solutions to 20
25abc, 28b
28cd, 31,32
34, 35, 37ab 38, 39
3, 5
8
1 1 sin2nn x
20. f (x) = 2 - -J2-----------------
7 òã z—4 n
n=1
L 4L TO cos[(2n - 1)-x/L]
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