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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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n=1
TO
y= E bn(m sin nt - n sin mt)/m(m2 - n2) + bm (sin mt - mt cos mt)/2m2, a
n= 1
n=m
?5- y = TO
n ซ2 - (2n - 1)2
11
sin(2n - 1)t sin at
2n 1 a
16. y = cos a
1 4 -TO
t + -------2 (1 — cos at) + —2 ^
2a n2
cos(2n - 1)n t - cos at
n2 TO1 (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 -TO 1 - cos(nn/2) nnx
14. f (x) = 7 + —^----------------2------cos'
4
n=1
2
4 -TO (nn/2) - sin(nn/2) nn x
f (x) = —2 > -----------------2----------sin -
n n=1 n2
2
1 2 TO
^ f(x) = 2 + -E
(-1)n
(2n - 1)n x
2 n i 2n - 1
n= 1
?TO 2 ( 2 nn \ nn x
16. f (x) = / — - cos nn + sin -3- I sin -
‘ J 11TT V nn "
n=1
17. f(x) = 1
n x—v
18. f (x) = -J2
7T Z 4
4 TO sin(2n - 1)x
n=1
2n 1
n=1
2 / nn 2nn
19. f (x) = ^ — ( cos + cos 2
cos nn sin
22 n n
33
nn
1
cos
nx
3
=m
Answers to Problems
727
See SSM for detailed solutions to 20
25abc, 28b
28cd, 31, 32
34, 35, 37ab 38, 39
3, 5
8
1 1 sin2nnx
20. f (x) = - - -J2--------------
7 7r z—4 11
n= 1
L 4L y cos[(2n - 1)nx/L]
21. f (x) = — +-------------------------------2------
" r2 (2n - 1)2
2
22. f{x) = ^ V sin(nnx/L)
n1
n 1 2n nn 4 / nn
23. (a) f(x) = - + - y —sin— + — (cos —
4 n , n 2 n2 \ 2
n=1
?y (-1)n
24. (a) f (x) = 2 > sin nx
/ n
25. (a) f(x) =
4n2n2(1 + cos nn) 16(1 - cos nn)
-J -J + -J -J
4 16 1 + 3 cos nn nn x
26. (a) f (x) = - + — y----2 cos -
3 n2 n=1 n2
3 6 1 - cos nn
27. (b) g(x) = - + — y 2 cos
/ TT' 1'
2 n 2 n_ 1
h( \ 6 "y 1 ? nn x
h(x) = — > — sin -
n “ n
n= 1
4 nn x
3
1 y
28. (b) g(x) = 4 + y
4cos (nn/2) + 2nn sin(nn/2) - 4 nn x
2
4 sin(nn/2) - 2nn cos(nn/2) nnx
h(x) = > ------------------------------------ sin -
22 n n
5 ^ 12cos nn + 4 nn x
29. (b) g(x) = -- + ?----------------^2--------cos.
h(x) = -2I]
n=1
y ซ2_2
2
1 n2n2(3 + 5cos nn) + 32(1 - cos nnx
n= 1
2
1 y
30. (b) g(x) = 4 + y
6n2n 2(2cosnn - 5) + 324(1 - cos nn) nn x
? , -----------------------;i—a--------------------cos------
4 n4 n4 3
h(x) =
4cosnn + 2 144 cos nn + 180
+ ^ ^
nn 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. [p(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—400n2t sin2nx — 2e—2500n2t sin5nx
8. u(x, t) = 2e—n t/16 sin(nx/2) — e-n t/4 sinnx + 4e—n t sin2nx
100 1 — cos nn „2„2,,1?nn nnx
9. u(x, t) = y---------------------------e-n n t/1600 sin -
n “ n
n= 1
40
nx
cos
2
2
22 i n
33
n n
728
Answers to Problems
See SSM for detailed solutions to 10
15abcd, 18a
18b, 19b, 20, 22
3, 7, 9abd
12abcd
14abc
10. ^, t) --tJ2
160 sin(nn/2) e-n2n2t/1600 sin ^
40
2
iuu x—v
11. u(x, t) --------------?
n=1
100 cOS(nn/4) - cOS(3nn/4) -n2n2t/1600 . nnx
n 40
80
12. u(x, t) - —^2
80 (-1T+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 2t,,nn nnx
18. u(x, t) - V--------------------------e-n n a t/400 sin-----
n n-1 n 20
(a) 35.91ฐC n (b)67.23ฐC (c) 99.96ฐC
19. (a) 76.73 sec (b) 152.56 sec (c) 1093.36 sec
21. (a) awxx - bwt + (c - b5)w - 0 (b) 5 - c/b if b - 0
22. ^" + /x2^ - 0, Y" + (k2 - x2)Y - 0, T + a2k2 T - 0
X'
23. r2#' + r# + (k2r2 - x2)? - 0, ฉ" + x2ฎ - 0, T + a2k2 T - 0
Section 10.6, page 588
1. u - 10 + 3x 2. u - 30 - 5x 3. u - 0
4. u - T 5. u - 0 6. u - T
7. u - T(1 + x)/(1 + I) 8. u - T(1 + L - x)/(1 + L)
70 cos nn + 50 nn x
9. (a) u(x, t) - 3x + V----------------------------e-0'86n n t/400 sin (d) 160.29 sec
nn 20
10. (a) /(x) - 2x, 0 <x < 50; f(x) - 200 - 2x, 50 < x < 100
OO
x X v 1 i/i„2„-2*/nnm2 nn x
(b) u(x, t) - 20------+ c e L ____
v > ( , ) 5 n 100
n1
800 nn 40
c - . . sin---------------------(d) u(50, t) ^ 10 as t ^ to; 3754 sec
n n2n2 2 nn
OO
X—v „2—2 qaa nn x
11. (a) u(x, t) - 30 - x + 22 cne-n n t/900 sin ,
n-1
60 2 2 c - —3—t\2(1 - cos nn) - n n (1 + cos nn)] n n
OO
2 x—' ™2_2^,2*/r2 nn x
12. (a) u(x, t) - - + > c e~ n a tlL cos —-—,
n “ n L
n-1
0, n odd;
-4/(n2 - 1)n, n even
(b) lim u(x, t) - 2/n
200 x—' ™2_2*/?4r?n nn x
13. (a) u(x, t) - — + ? cne-n n t/6400 cos —,
n- 1
160
cn --------(3 + cos nn)
n 3n n2
(c) 200/9 (d) 1543 sec
.-“i r- OO
25 x—' r,2—.2f/Qnn nn x
14. (a) u(x, t) - — + V c e-n n t/900 cos-------------------------,
y ^ ( , ) 6 ^ n 30 ,
n=1
50 / nn nn
c = — sin sin —
n nn \ 3 6
Answers to Problems
729
See SSM for detailed solutions to 15a
15b, 19
1a
1bce, 6ab
6c, 9
15. (b) u(x, t) = J2<
2n2a2t/4L2 ? _ (2n - 1)nx
-(2n-1)2n a t/4L
2 [L ^ ? (2n - 1)nx
C = — I I (x) sin----------;------- dx
n L JQ () 2L
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