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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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23. r2R" + ãß+ (X2r2 - x2)R = 0, ©" + x2© = 0, T + a2X2 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 + L) 8. u = T(1 + L - x)/(1 + L)
70 cos nn + 50 ÏîË„2„.2,ìò nnx
9. (a) u(x, t) = 3x + V-------------------e-0'86n n ‘/400 sin------------------- (d) 160.29 sec
^"1 nn 20
10. (a) f(x) = 2x, 0 < x < 50; f(x) = 200 - 2x, 50 < x < 100
OO
x x v 1 . nn x
(b) u(x, t) = 20-----------+ c e 1' ____
( 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—' ë2ò7-2^þïï 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—' r\2tt2rJ211 T ^ 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
t^TO
200 x—' nn x
13. (a) u(x, t) = — + ? cne-n n t/6400 cos —,
Ï= 1
160
cn =--------(3 + cos nn)
n 3n n2
(c) 200/9 (d) 1543 sec
ë ? OO
25 x—' ö2„-2*/îïï nn x
14. (a) u(x, t) = — + V c e-n n t/900 cos-----------------------------,
y ^ ( , ) 6 ^ n 30 ,
Ï= 1
50 / nn nn
c = — sin------------------sin —
n nn V 3 6
730
See SSM for detailed solutions to 10a 10bc, 13, 14
16,17abc 18abc, 24
1abc
1d
2, 3a, 4
5, 7
8abc
8 -TO 1
10. (a) u(x, t) = - ~-------7
— 2n— 1
8 1 (2n — 1)— (2n — 1)— (2n — 1)—x (2n — 1)— at
sin-------------- sin------------- sin---------------- cos ¦
— 2n - 1 4 2L 2L 2L
n=1
512 (2n — 1)— + 3cos n— (2n — 1)—x (2n — 1)—at
11. (a) u(x, t) = ~Y ---------------------------3--------sin ---------------cos-

n=1
(2n - 1)4 2L 2L
14. ô (x + at) represents a wave moving in the negative x direction with speed a > 0.
15. (a) 248 ft/sec (b) 49.6—n rad/sec (c) Frequencies increase; modes are unchanged.
16. u(x, t) = ^^ cn cos Pnt sin(n— x/L), â2 = (n— a/L)2 + à2
n=1 -L
2 f1
cn = — I f (x) sin(n— x/L) dx
n L J0
22. r2R" + rR + (X2r2 - /j2) R = 0, ®" + /ë2® = 0, T" + X2a2 T = 0
Section 10.8, page 611
TO n— x n— y 2/a a n— x
° (a) u(x, y) = > c sm-------------sinh--------, c = --------------- g(x) sin----dx
~° a a sinh(n— b/a) j0 a
4a -TO 1 sin(n—/2) n— x n— y
(b) u(x, y) = -^r Ó —ã--------------------sin---------sinh-----
—2 nr! n sinh(n— b/a) a a
Æ . n— x n—(b- y~) 2/a fa . n— x
2. u(x, y) = > c sin---------------sinh----------------------------------, c = - I h(x) sin------ dx
¦^-0 n a a n sinh(n— b/a) J0 a
m n— x n— y ïË n— x n—(b- y)
3. (a) u(x, y) = ó c'n ) sinh----------sin---------+ > c{2) sin-----sinh----------------,
*—! n b b *—! n a a
n=1 n=1
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