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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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31-34. In all cases solution is y -I G(x, s) /(s) ds, where G(x, s) is given below.
0
31. G(x, S) - )1 - x, 0 < s < x 32. G(x, S) - js(2 - x)/2, 0 < s < x
[1 - s, x < s < 1 lx(2 - s)/2, x < s < 1
cos
734
Answers to Problems
See SSM for detailed solutions to 33
2abc
4ab
1abc
2
4, 5ab, 6 7b, 9a, 10
, I cos s sin( 1 — x)/ cos1, 0< s<x
33. G(x, s) = { . ,, \ , ’ _ _ ,
[sm(1 - s) cos x/ cos1, x < s < 1
34. G(x, s) = js, 0 < s < x
x, x < s < 1
Section 11.4, page 661
?x. c p 1 I ? 1
1. y =J2Jo(^/ãnx), cn = f(x)dx xJ0(VKx) dx,
n=i An - G J0 /70
kn - G
JTn satisfies J0 (\/X) = 0 c x c 2 (c)y=-G + Er-G.J^n*);
n=1 n -
c0 = 2 f0 f(x) dx; cn = ^ f(x) J0(/Tnx) dx j^ xJl(JTnx) dx, n = 1, 2,...; yT satisfies J0 (vT) = 0
3. (d) an = / xJ, (/^^x) f (x) dx j xJ, (JTnx) dx
x c /1 If 1
(e) y = 1] T^G J (^‘nx), cn = f(x) J ^^rnx) dx xJk(^nx) dx
n= 1 n G * 0 / «/0
4. (b) y = f] P2n-1(x) cn = / f (x) P2n- 1(x) dx / f P2n- 1(x) dx
n=1 kn - G 70 /70
n=1 Tn - G
Section 11.5, page 666
1. (b) u(S, 2) = f(S + 1), u(S, 0) = 0, 0 < S < 2
u(0, n) = u(2, n) = 0, 0 < n < 2
1 /*1 / /*1
2. u(r, 0 = XI knJ0 (knr) sin knaf’ kn = 1 rJ0(k„r)g(r) dr/ rJ0(k„r) dr
n= 1 kna J0 / -f0
3. Superpose the solution of Problem 2 and the solution [Eq. (21)] of the example in the te
<x ? 1 j ? 1
6. u(r, z) = V cne-knZJ0(knr), cn = rJ0(knr) f (r) dr / rj0f(knr) dr,
n 1 0 0
0
and kn satisfies J0(k) = 0.
to
7. (b) v(r, 9) = 1 c0 J0(kr) + Y Jm(kr)(bm sinm9 + cm cosm6)
m=1
2n
bm = _ j. /f ^ j f(0) sinm0 d0; m = 1, 2,...
m(kc) L
1 f2n
Jkc)j0
c = —----------- I f (9) cos m9 d9; m = 0, 1, 2,...
m * Jm (kc)Jo
8. cn = jT rf (r) J0(knr) dr j rJ^(knr) dr
TO /> 1 j /. 1
10. u(p, s) = ^ cnpnPn(s), where cn = J f (arccos s)Pn(s) ds j J f^is) ds;
Pn is the nth Legendre polynomial and s = cos $.
Answers to Problems
735
See SSM for detailed solutions to 2abc, 4a
5, 7bed, 8 9abcde, 10, 12
Section 11.6, page 675
1. n = 21
2. (a) bm = (— 1)m+l\fl/mn (c) n = 20
3. (a) bm = 2\fl(1 — cosmn)/m3n3 (c) n = 1
7. (a) 70(x) = 1 (b) f1(x) = V3(1 — 2x) (c) f2(x) = V5(—1 + 6x — 6x2)
(d) g0(x) = 1, g1(x) = 2x — 1, g2(x) = 6x2 — 6x + 1
8. P0(x) = 1, P1(x) = x, 7^(x) = (3x2 — 1)/2, P^(x) = (5x3 — 3x)/2
INDEX
A
Abel, Niels Henrik, 149, 216 Abel’s formula, 149, 167, 168, 213,228 for systems of equations, 370 Acceleration of convergence, 564 Adams, John Couch, 439 Adams-Bashforth formula, fourth order, 440 second order, 440 Adams-Moulton formula, fourth order, 441 second order, 441 Adaptive numerical method, 427, 433
Adjoint, differential equation,
146 matrix, 348 Airy, George Biddell, 243 Airy equation, 147, 243-247,
252, 260, 291, 309 Almost linear systems, 479-491 Amplitude, of simple harmonic motion, 190 Amplitude modulation, 202 Analytic function, 235, 250 Angular momentum, principle of, 473-474 Annihilators, method of, 225-226 Asymptotic stability, see Stability Augmented matrix, 358
Autonomous, equation, 74 system, 471-472
B
Backward differentiation formulas, 443-444 Backward Euler formula, 422-424 Basin of attraction, 487, 498, 516-519 Beats, 201-202, 323-324 Bendixson, Ivar Otto, 525 Bernoulli, Daniel, 25, 87, 88,
573, 591 Bernoulli, Jakob, 24, 63, 73, 336 Bernoulli, Johann, 24, 63 Bernoulli equation, 73 Bessel, Friedrich Wilhelm, 280 Bessel equation of order: k, 662
nu, 147, 152, 238, 256, 259, 260, 280, 290, 627, 657 one, 272, 287-289, 290 one-half, 285-286, 289 zero, 272, 280-284, 290, 308, 613, 658, 665 Bessel functions, 25
J0(x), 272, 281, 289, 308, 309, 658, 660-661, 662, 665, 668
asymptotic approximation to, 284 Laplace transform of, 308 zeros of, 291, 658, 665 J1(x), 272, 287, 289 J1/2(x), 285 J-1/2(x), 286
orthogonality of, 291, 661, 662 Y0(x), 283, 658, 665 asymptotic approximation to, 284 Y1(x), 289 Bessel inequality, 676 Bessel series expansion, 661, 666 Bifurcation (points), 88, 89, 122, 383, 480, 502, 533 Boundary conditions: 542 for elastic string, 591, 600 for heat equation, 574, 582, 584, 616 for Laplace’s equation, 605 nonhomogeneous, 582-584, 650
periodic, 638, 674 separated, 630 time dependent, 650 Boundary layer, 450 Boundary value problems: heat conduction equation, 573-590, 614-617, 646-649
737
738
Index
homogeneous, 542-544, 629-641 Laplace’s equation, 604-613 nonhomogeneous, 542-543, 641-645 self-adjoint, 637-639, 660 singular, 656-663 Sturm-Liouville, 629-630 two-point, 541-547, 623 wave equation, 591-604, 617-619, 653, 664-666 see also Homogeneous
boundary value problems; Nonhomogeneous boundary value problems Brachistochrone, 24, 63 Buckling of elastic column,
640-641
C
Capacitance, 195, 196 Cardano, Girolamo, 216 Cayley, Arthur, 348 Center, 387, 467, 479, 490 Change of independent variable,
159-160, 291 for Euler equation, 160, 264, 266
Chaotic solution, of logistic
difference equation, 122, 126
of Lorenz equations, 536 Characteristic equation, 132, 214, 303
complex roots, 153, 217 real and equal roots, 161, 218 real and unequal roots, 132,
215
Characteristic polynomial, 214, 303
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