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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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(1) 2/b fb n— y (2) 2/a fa n— x
c(1) = --------------- I f (y) sin-------- dy, c(2) = ------------------- I h(x) sin-------- dx
sinh(n— a/b) J0 b sinh(n— b/a) J0 a
(1) 2 (2) 2 (n2—2 - 2) cos n— + 2
n— sinh(n— a/b) n n3—3 sinh(n— b/a)
c_0 TO
2
(b) c(1) = ---------- --------- c(2) = -
n— sinh( r,-o/h), n „3^3
5. u(r, â) = — + Xr-n(cn cos n6 + kn sinn0);
2 n=1
„n f2— 0n f2—
a“ f an f
c = — I f (â) cos ne de, k = — I f (â) sinne de
n -Jo n — Jo
n 2
‘ sin ne, c =
, n —a“ J0
n=1 ° 0
4 cos n— 1
(b) cn = ----------f-------
nn
— J0 — J0
TO 2 f
6. (a) u(r,e) ^Ó c rn sin ne, c =—- I f (â) sin ne de
n 1 n n —an 0
— a n
TO n—â Ãà n—â
7. u(r, â) = V c rn-/à sin-, c = (2/à)a-n-/à f (â) sin---de
n 1 n à n 0 à
n=1
n— y/a n— x 2 fa n— x
8. (a) u(x, y) =} c e-n—y/a sin-------------------, c = - f (x) sin------------- dx
n a n a Jn a
4a2
(b) cn = —ç—ç (1 — cos n—) (c) y0 = 6.6315
b
n— x n—y 2/n— f n—y
00 (b> u(x,y) = c0 + n=° cn cosh — coS —, cn = sinh-— a/b) ³ f (y) cos ~ dy
11. u(r,e) = c0 + E rn (cn cos ne + kn sin ne),
n=1
1 f2— 1 f2—
c = ---------— g(e) cos ne de, k = -----------------------— g(e) sin nâ de;
n— an 1 J0 n— an 1 J0
r 2—
ition is
0
necessary condition is g(e) de = 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) = Ó 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---------------,
n 2b 2b
Ï= 1
2/b f b (2n — 1)n y
f
0
f (y) sin-------------------- dy
n sinh[(2n — 1)na/2b] J0 2b
32b2
(b) cn =----------773 3----------
(2n — 1) n3 sinh[(2n — 1)na/2b] cos
co y 1 nn x . u nn y
2 fa 2/a fa nn x
co = Til g(x 1 dx cn = sinh(nn b/a) ³ g(x) COs — dx
nn
14. (a) u(x, y) = ——+ Ó c cos-sinh-
2 •“ a a
n= 1
24a4(1 + cos nn) b\ 30/’ n n4n4 sinh(nn b/a)
(b) co = 7 1 + ™
Section 11.1, page 626
1. Homogeneous 2. Nonhomogeneous 3. Nonhomogeneous
4. Homogeneous 5. Nonhomogeneous 6. Homogeneous
7. ôï(x) = sin yX-x, where satisfies = — tan n;
X1 = 0.6204, X2 = 2.7943, Xn = (2n — 1)2/4 for large n
8. ôï(x) = cos x, where satisfies *JX = cot vX
X1 = 0.7402, X2 = 11.7349, Xn = (n — 1)2n2 for large n
9. ôï(x) = sin tJX~x + tJX~cos ^/X-x, where ^/X- satisfies
(X — 1) smVI — 2\pX cos \/X = 0;
X1 = 1.7071, X2 = 13.4924, Xn = (n — 1)2n2 for large n
10. X0 = 0; ô^) = 1 — x
For n = 1, 2, 3,...,ô (x) = sin ä x — ä cos ä x and X = — ä^, where ä satisfies ä = tan ä.
X1 = —20.1907, X2 = —59.6795, Xn = — (2n + 1)2n2/4 for large n
2
12. /u(x) = e x 13. p,(x) = 1 /x
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