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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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21. f (x) = — +-----.--------------------------------
" r2 j=1 (2n - 1)2
2
22. f(x) = ^ Ó sin(n-x/L)
n1
— 1 Xãî 2— n— 4 / n-
23 (a) f (x) = T + - Ó — sin T" + T (cos “T
4 - n 2 n2 2
n=1 L
-TO (-1)n
24. (a) f (x) = 2 > -------------------sin nx
' 11
25. (a) f(x) =
4n2—2(1 + cos n—) 16(1 — cos n—)
³ ³ + ³ ³
4 16 1 + 3 cos n— n— x
26. (a) f (x) = - + — Ó---------2-------cos ¦
3 — 7=1 n2
3 6 1 — cos n—
27. (b) g(x) = - + — > ----------------------2--------cos
/ ÒÃË Z----4 11
2 —2 n 1 ... 6 -TO 1 ¦ n— x
h(x) = — > — sin ¦
—n
n=1
4 n— x
3
1 TO
28. (b) g(x) = 4 + X
4cos (n—/2) + 2n— sin (n—/2) — 4 n— x
------------------------------------cos-------
n2—2 2
4 sin(n—/2) — 2n— cos(n—/2) n— x
h(x) = > ------------------------------------sin ¦
22
n2—2
5 ^ 12cos n— + 4 n— x
29. (b) g(x) = -- + ?----------------^2-------------cos.
h(x) = -21]
n=1
TO ~2_2
2
1 n — 2(3 + 5cosn—) + 32(1 — cosn— n—x
n=1
2
1 TO
30. (b) g(x) = 4 + J2
6n2—2(2cosn— — 5) + 324(1 — cos n—) n— x
¦ , -------------------ë—ë-----------------cos----
4 n4—4 3
h(x) =
4cos n— + 2 144 cos n— + 180
+ ç ç
n— x
40. Extend f(x) antisymmetrically into (L, 2L]; that is, so that f(2L — x) = — f(x) for
0 < x < L. Then extend this function as an even function into (—2L, 0).
Section 10.5, 579
1. xX"- XX = 0, T + XT = 0 2. X"- XxX = 0, T + XtT = 0
3. X" - X(X' + X) = 0, T + XT = 0 4. [jo(x)X']' + Xr(x)X = 0, T" + XT = 0
5. Not separable 6. X" + (x + X)X = 0, Y" — XY = 0
7. u(x, t) = e-400—2f sin2— x - 2e-2500—2f sin 5—x
8. u(x, t) = 2e-— f/16 sin(— x/2) — e-— f/4 sin—x + 4e-— f sin2—x
100 1 — cos n— „2„2,,1éïï n— x
9. u(x, t) = -Ó-e-n — f/1600 sin -
—n
n=1
40
nx
cos
2
2
33
n —
728
See SSM for detailed solutions to 10
15abcd, 18a
18b, 19b, 20, 22
3, 7, 9abd
12abcd
14abc
1UU x—v
10. u(x, t) = -tJ2
160 ^ sin(nn/2) e_n2n2t/1600 sin nnx
40
2
n=1
100 cos(nn/4) - cos(3nn/4) -n2n2^1600 ¦ nnx
n 40
80
12. u(x, t) = —^2
80 (-1_)Ï+1 e-n2n21/1600 sin nn x
n 40
13. t = 5, n = 16; t = 20, n = 8; t = 80, n = 4
14. (d) t = 673.35 15. (d) t = 451.60 16. (d) t = 617.17
17. (b) t = 5, x = 33.20; t = 10, x = 31.13; t = 20, x = 28.62; t = 40, x = 25.73;
t = 100, x = 21.95; t = 200, x = 20.31
(e) t = 524.81
200 1 — cos nn „2„2,*2,ìò nn x
18. u(x, t) = ------V-----------------e-n n a 1 /400 sin-----
n n 20
(a) 35.91°C Ï (b)67.23°C (c) 99.96°C
19. (a) 76.73 sec (b) 152.56 sec (c) 1093.36 sec
21. (a) awxx - bwt + (c - b5)w = 0 (b) & = c/b if b = 0
22. ^" + /X2^ = 0, Ó" + (X2 - /ë2)Ó = 0, T + a2X2T = 0
X
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