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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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n= 1 Xna "0 / "0
3. Superpose the solution of Problem 2 and the solution [Eq. (21)] of the example in the te
TO ? 1 j ? 1
6. u(r, z) = V cne-X-ZJ0(Xnr), cn = rJ0(Xnr) f (r) dr / rJ0;(Xnr) dr,
n 1 0 0
0
and X- satisfies J0X = 0.
TO
7. (b) v(r, â) = 1 c0 J0(kr) + Jm(kr)(bm sinme + cm cosøâ)
m=1
2n
bm = _ ã ÷ j f(e) sin me de; m = 1, 2,...
m(kc) Ë
1 Ã2n
Jkc)j0
c = —---------- I f (â) cos me de; m = 0, 1, 2,...
m n Jm (kc)j0 K ’ ’ ’ ’ ’
8. cn = jT rf (r) J0(Xnr) dr j rJ^(Xnr) dr
TO /> 1 j /> 1
10. u(p, s) = X cnpnP-(s), where cn = J f (arccos s)P-(s) d^ó J P’2(s) ds;
Pn is the nth Legendre polynomial and s = cos ô.
735
See SSM for detailed solutions to 2abc, 4a
5, 7bcd, 8 9abcde, 10, 12
Section 11.6, page 675
1. n = 21
2. (a) bm = (— ³)m+l\[0/mn (c) n = 20
3. (a) bm = 2^(1 — cosmn)/m3n3 (c) n = 1
7. (a) /0(x) = 1 (b) f1(x) = V3(1 - 2x) (c) f2(x) = \/5(-1 + 6x - 6x2)
(d) g0(x) = 1, g1(x) = 2x - 1, g2(x) = 6x2 6x + 1
8. P0(x) = 1, P1(x) = x, P2(x) = (3x2 — 1)/2, ^3(x) = (5 õ3 — 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
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 Jy2(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
Chebyshev, Pafnuty L., 253, 511 Chebyshev equation, 253, 271, 627, 663 Chebyshev polynomials, 253, 663 Chemical reactions, 89 Collocation, method of, 670 Competing species, 491-503 Complementary solution, 170 Complete set of functions, 672
Complex exponentials, 153-155, 219
Compound interest, 51-54 Computer use in differential equations, 21 Conjugate matrix, 348 Continuous spectrum, 660 Convergence:
of an improper integral, 294 of a numerical approximation, 424
of a power series, 232 Convergence in mean, 672 Converging solutions, 4, 12, 102 Convolution integral, 185, 330-337 Laplace transform of, 330-333 Cosine series, 566 Critical amplitude, 82 Critical damping, 193 Critical point:
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