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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
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n 1
2L
16. (a) u(x, t)
n 1
120
-(2n-1)2n2t/3600 sin (2n 1)nx
60 ,
r[2cosnn + (2n - 1)n]
n (2n - 1)2n2
(c) xm increases from x = 0 and reaches x = 30 when t = 104.4.
17. (a) u(x, t) = 40 + cne-(2n-1)2n2t/3600 :
40
n (2n - 1)2n
19. u(x) =
Tx-
L + ( L/a) - 1.
Lx
1
a s +(L / a) - 1 _
20. (e) un(x, t) = e-An“2t sinknx
(2n - 1)nx 60 ,
: [6 cos nn - (2n - 1)n ]
0 < x < a,
a < x < L,
where S = k2 A2/k1 A1
Section 10.7, page 600
1. (a) u(x,
2. (a) u(x,
3. (a) u ( x,
4. (a) u(x,
5. (a) u(x,
6. (a) u(x,
7. (a) u(x,
8. (a) u(x,
8 1 nn nn x nn at
t) = —t y 3 sin — sin —— cos —;—
; n2 n2 2 L L
8 1 ( nn 3nn\ nnx
t) = — > — sin + sin sin ——
7 2 „2 \ 4 4 / L
n=1
nn x nn at
L C0* —
32 2 + cos nn nn x nn at
t) = —3 y ---------------3------sin —— cos —-—
n n=1 n L L
4 sin(nn/2) sin(nn/L) nn x nn at
t) = — y ------------------------------------ sin —— cos --;---
n *—! n L L
n=1
8L ^ 1 nn nn x nn at
t) = —3 y —3 sin —— sin —— sin -
o nr ' J rP
n= 1
2
L
L
8L sin(nn/4) + sin(3nn/4) nn x nn at
t) = —3 y --------------------T--------------sin —— sin -
an n
L
L
3 2L cos nn + 2 nn x nn at
t) = —4 L, 4 sin sin '
an nL n
4L sin(nn/2) sin(nn/L) nn x nn at
t) = —3 > ------------------;-----------sin —— sin -
an2 n2
LL
in
~L
L
(2n - 1)nx (2n - 1)nat
9. u(x, t) = y_^ cn sin — cos -
cn L
0
n 1 2L
n= 1
L er ^ ? (2n - 1)nx J
i (x) sin ;---------------- dx
2L
2L
a
2
730
See SSM for detailed solutions to 10a 10bc, 13, 14
16, 17abc 18abc, 24
1abc
Id
2, 3a, 4
5, 7
8abc
8 ^ 1
10. (a) u(x, t) = ~------7
n *—! 2n— 1
8 1 (2n — 1)n (2n — 1)n (2n — 1)nx (2n — 1)nat
sin------------- sin------------ sin--------------- COS ?
n ^ 2n — 1 4 2L 2L 2L
n=1
512 (2n — 1)n + 3cos nn (2n — 1)nx (2n — 1)nat
11. (a) u(x, t) = —E ------------------------; sin -r2- cos-
n
n=1
(2n — 1)4 2L 2L
14. 0(x + at) represents a wave moving in the negative x direction with speed a > 0.
15. (a) 248 ft/sec (b) 49.6nn rad/sec (c) Frequencies increase; modes are unchanged.
16. u(x, t) = ^^ cn cos ?nt sin(nnx/L), ?2n = (nna/L)2 + a2
n= 1 -L
2 f1
cn = — I f (x) sin(nn x/L) dx
n L Jo
22. r2R" + rR + (k2r2 - /j2)R = 0, ®" + j2© = 0, T" + k2a2 T = 0
Section 10.8, page 611
nn x nn y 2/a fa nn x
1. (a) u(x, y) = > c sin sinh , c = ------------------------------------------------------------------ g(x) sin dx
a a sinh(nn b/a) J0 a
4a 1 sin(nn/2) nn x nn y
(b) u(x, y) = -^r y —r------------------------------sin---------------------------sinh-
n n=1 n sinh(nn b/a) a a
^ . nnx nn(b - y~) 2/a fa . nn x
2. u(x, y) = > c sin sinh , c = - I h(x) sin dx
TI n a a n sinh(nn b/a) J0 a
m nnx nn y nnx nn(b— y)
3. (a) u(x, y) = } cl sinh sin + > c2) sin sinh-------------------------------------------,
*—! n b b n a a
n= 1
(1) 2/b fb nn y (2) 2/a fa nn x
c“) = I f (y) sin dy, c“2) = --- I h(x) sin dx
sinh(nn a/b) J0 b sinh(nn b/a) J0 a
(1) 2 (2) 2 (n2n2 — 2) cos nn + 2
nn sinh(nna/b) n n3n3 sinh(nn b/a)
c_0 ™
2
(b) cw = ---------------------- c(2) = -
nn sinh( n^o/h)’ n „3_3
5. u(r, 6) = — + ^r—n(cn cos nd + kn sinn0);
2 n=1
„n f2n 0n f2n
a“ f an f
c = — I f (6) cos n6 d6, k = — I f (6) sinn6 d6
n -Jo n n J0
„ 2 rn
‘ sin n6, c =
, n na“ J0
n=1 0
4 cos nn + 1
(b) cn = —-----------------3-----
nn
n J0 n J0
^ 2 f
6. (a) u(r,6) = > c rn sinn6, c =—- I f (6) sinn6 d6
TI n n n an J0
n a n
nn6 Ca nn6
7. u(r, 6) = V c rm/a sin , c = (2/a)a—nn/a f (6) sin-------d6
TI n a n J0 a
n=1
y/a nnx 2 fa nnx
8. (a) u(x, y) = > c e—nny/a sin , c = - f (x) sin dx
n=1 n a n a 0 a
4a2
(b) cn = —3—3 (1 — cos nn) (c) y0 = 6.6315
b
nn x nny 2/nn f nny
10' (b) u(x’y' = c0 + ^ cn cosh ~ C°S — ’ cn = sinh(nn a/b) l f (y) cos T" dy
11. u(r,6) = c0 + E rn (cn cos n6 + kn sin n6),
n=1
1 f2n 1 f2n
c = ------------— g(6) cos n6 d6, k = ----------— g(6) sinn6 d6;
nn an 1 J0 nn an 1 J0
c 2n
ition is
0
necessary condition is g(6) d6 = 0.
731
See SSM for detailed solutions to 13a
CHAPTER 11
2, 4, 5, 9
10, 11ab
13, 18a, 20
22abcd, 23
nn x nny 2/a fa nn x
12. (a) u(x, y) = y c sin cosh , c = -------------------------------- I g(x) sin dx
W 7 ^ n a a n cosh(nn b/a)J0S a
n=1
4a sin(nn/2)
(b) cn = —------------------------
n n cosh(nn b/a)
(2n — 1)n x (2n — 1)n y
13. (a) u(x, y) = c sinh---------------------------------------sin--------------------------------------,
w ( ?jv n 2b 2b
n= 1
2/b f b (2n — 1)n y
f
Jo
f (y) sin--------------------- dy
n sinh[(2n — 1)na/2b] J0 2b
32b2
(b) cn =--------773 3-----------
(2n — 1)3n3 sinh[(2n — 1)na/2b] cos
co y I nn x . u nn y
2 fa 2/a fa nn x
co = TbJo g(x' dx? cn = sinh(nnb/a) ]„ g(x) COs ~ dx
V x—V IUI A un y
14. (a) u(x? y) = ——+ y c cos sinh------------------------------
2 “ a a
n= 1
24a4(1 + cos nn) b\ 30/’ n n4n4 sinh(nn b/a)
(b) co = 7 1 + 77
Section 11.1, page 626
1. Homogeneous 2. Nonhomogeneous 3. Nonhomogeneous
4. Homogeneous 5. Nonhomogeneous 6. Homogeneous
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