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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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14. fx(x) = e—x 15. fx(x) = (1 — x2)—1/2
16. a2^' + (X — *)^ = 0, T" + cT' + XT = 0
17. (a) s (x) = ex (b) Xn = n2n2, ôï (x) = ex sin nn x; n = 1, 2, 3,...
18. Positive eigenvaluesX = Xn, where ,jX~n satisfies \/X = § tan3\/XL; corresponding eigenfunctions are ôï (x) = e—2x sin 3^/X-x. If L = 2 ,X0 = 0 is eigenvalue, ô0(x) = xe—2x
is eigenfunction; if L = 2 ,X = 0 is not eigenvalue. If L < i, there are no negative eigenvalues; if L > 2, there is one negative eigenvalue X = — /ë2, where ä is a root of ä = 3 tanh 3 äL; corresponding eigenfunction is ô_ 1 (x) = e—2x sinh 3^x.
19. No real eigenvalues.
20. Only eigenvalue is X = 0; eigenfunction is ô (x) = x — 1.
21. (a) 2 sin VX — \/X cosVX = 0; X1 = 18.2738, X2 = 57.7075 (b) 2 sinh^/ä — „óä coshy^ = 0, ä = — X; X_ 1 = — 3.6673
732
See SSM for 24. (a) Xn = /ë4ï, where in is a root of sin /.iL sinh /.iL = 0, hence Xn = (nn/L)4;
detailed solutions X1 = 97.409/L4, Ôï(x) = sin(nnx/L)
to 24 b (b) Xn = /ë4ï, where in is a root of sin iL cosh iL — cos ë L sinh ë L = 0;
, sin i x sinh i L — sin i L sinh i x
X, = 237.72/L , ôï = -----------^^
1 sinh inL
ôï (x) =
(c) Xn = /ë4ï, where in is a root of 1 + cosh iL cos iL = 0; X1 = 12.362/L4,
[(sin inx — sinh inx)(cos inL + cosh inL) + (sin inL + sinh inL )(cosh inx — cos inx)]
cos inL + cosh inL
25. (c) ôï(x) = sinyX"nx, where Xn satisfies cos .JX, L — ó^X, L sin^JX, L = 0;
X1 = 1.1597/L2, X2 = 13.276/L2
Section 11.2, page 639
1. Ôï(x) = \/^³ï(ï — 2)nx; n = 1, 2,...
1,3,5 2. ôï(x) = ^/0cos(n — 1 )nx; n = 1, 2,...
3. ô,^) = 1, ôï(x) = V^cosnnx; n = 1, 2,...
V! cos V^n x - - -
4. ôï(x) = (1 + sin2 yX^)1/2 , where Xn satisfies coVXn -VXn sinVXn = 0
2\/0
5. ô (x) = v/0 e* sin nnx; n = 1, 2,... 6. a = -------------; n = 1, 2,...
n ^ n 1 n (2n - 1)n
^VK—1)n—1
7,10,14,17,21a 7. an = ^------:òòã; n = 12-..
(2n - 1) n 2 /2
8. a =------------{1 — cos[(2n — 1)n/4]}; n = 1, 2,...
n (2n - 1 )n
2V0sin(n - 1 )(n/2)
9. an =------------1 2 2--------; n = 1, 2,...
n (n - 1 )2n 2
In Problems 10 through 13, àï = (1 + sin2 ^X,)1/2 and cos óÕ, — ^X, sin óÕ, = 0.
ÓÐð cos ^X~ - 1) _
Xnàn ’
sin^/2) _
10. a -Ä sin Jx~ : 1,2,... 11.
---7=^---; n =
n jXn àï
12. a Ë(1 - cos^X) n = 1, 2,... 13.
n X à
nn
21. (a) If a2 = 0 or b2 = 0, then the corresponding b
25. (a) X1 = n /L ; ô1 (x) = sin(nx/L)
(b) X , = (4.4934)2/L2; ô1(x) = sin^X x-
(c) X1 = (2n )2/L2; ô1 (x) = 1 --- cos(2n x / L)
26. X1 = n 2/4L2; ô1(x) = : 1 --- cos(n x /2 L)
, n = 1, 2,..
Xn àï
n = 1, 2,...
21 be, 23abc, 24 25bc
Section 11.3, page 651
1 ^ (-1)n+1sinnnx ^ (-1)n+1sin(n - 1 )nx
1 y = 2Ú („2n2 — .w 2 y = 2L
^ (n2n2 - 2)nn ^ [(n - 2)2n2 - 2](n - 1 )2n2
n
733
See SSM for detailed solutions to 3, 5, 8
10, 11, 12, 14, 18
19, 22
24,28a
28bcd
30bcd
30e
cos(2n — 1)—x
TO
3. y=-------------4^
4 -To [(2n — 1)2—2 — 2](2n — 1)2—2
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