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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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approached by trajectory, 467 center of linear system, 387, 467, 479, 490 definition of, 460, 472 for first order equation, 77 improper node, 404, 462-464 isolated, 481
node of linear system, 378, 390, 460-461 nonisolated, 470 proper node of linear system, 462
saddle point of linear system, 375, 389, 461-462 spiral point of linear system, 386, 390, 464-466 stability of, see Stability of critical point Cycloid, 64, 337
D
D’Alembert, Jean, 161, 573, 591, 602
Dal Ferro, Scipione, 216 Damping force, 187-188, 473 Decay, radioactive, 16, 59 Degenerate node, see Improper node
Diagonalization, of matrices,
397-398
of homogeneous systems,
398-400
of nonhomogeneous systems, 411-413 Difference equation, 115-126 equilibrium solution of, 116 first order, 115-126 initial condition for, 116 iteration of, 116 linear, 115, 116-118 logistic, 118-126
chaotic solutions of, 122,
126
nonlinear, 115, 118-126 solution of, 115 Differential operator, 138, 630 Diffusion equation, see Heat conduction equation Dimensionless variables, 580,
601
Dirac, Paul A. M., 326 Dirac delta function, 326, 656 Laplace transform of, 326-327 Direction field, for first order equations, 3, 5 for systems, 373 Dirichlet, Peter Gustav Lejeune, 605
Dirichlet problem, 605 for circle, 609-611, 612 for rectangle, 606-608, 611-612 for sector, 612 for semicircle, 612 for semi-infinite strip, 612 Discontinuous forcing function, 74,317-324 Diverging solutions, 6, 10, 102 Drag, 2, 8
Duffing equation, 478 E
Eigenfunctions, 545, 623 normalized, 633 orthogonality of, 632, 660 series of, 635-636, 642, 660-661, 672
Index
739
Eigenvalues of linear
homogeneous boundary value problems, 545,
623
of Sturm-Liouville problem, existence, 631 real, 547, 631 simple, 633 when positive, 640 Eigenvalues of matrix, 362-366, 401-402 multiplicity, 364 real, 365 simple, 364 Eigenvectors of matrix, 362-366, 401-402 generalized, 405, 409 linear independence of, 364, 365
normalized, 364 orthogonality of, 366 Elastic bar, vibrations of,
628-629 Elastic membrane, vibrations of, 604, 664-666 Elastic string: boundary value problems for, 591-604 derivation of wave equation, 617-619 free at one end, 600 general problem, 598-600 infinite length, 602-603 justification of solution, 596-598, 603-604 natural frequencies of, 594 natural modes of, 594 propagation of discontinuities in initial conditions, 598 wavelength of, 594 Electrical networks, 16, 195-196, 306-307, 340, 345-346 Elliptic integral, 491 Environmental carrying capacity, 78
Epidemics, 87-88 Equilibrium solution, 4, 6, 12, 76, 116,460 see also Critical point
Error:
for Adams-Bashforth formula,
440
for Adams-Moulton formula,
441
effect of step size, 426, 445-447 for Euler method, 425-427, 428
for Fourier series, 554-555, 561-562 global truncation, 424, 426, 445-447 for improved Euler method, 431,434-435 local truncation, 424 mean square, 671 for modified Euler method, 435 round-off, 424, 429, 445-447 for Runge-Kutta method, 436 Escape velocity, 56-57 Euler, Leonhard, 24, 25, 97, 571,
573, 591,604 Euler equation, 160, 169,
260-267, 382, 391, 408, 610
change of independent
variable, 160, 264, 266 Euler formula for exp (it), 154 Euler-Fourier formulas, 550-551 Euler-Mascheroni constant, 283 Euler method, 96-105, 419-429, 455-457 convergence of, 104-105 global truncation error, 426 local truncation error, 425-427 Even functions, 564 Even periodic extension, 568 Exact equations, 89-93, 146 necessary and sufficient
condition for existence of solution, 25, 91 for second order equations, 146 Exchange of stability, 120 Existence and uniqueness theorems, 20, 21 for first order equations, 66,
106
for first order linear equations, 65
for nth order linear equations, 210
proof of, for y = f (t, y), 106-113 for second order linear equations, 138 for series solution of second order linear equations, 250,277-278 for systems of first order equations, 343
Exponential growth, 75
Exponential matrix, 396-397, 400
Exponential order, functions of, 296
Exponents at the singularity, 269, 274
F
Falling object problem, 2, 3, 12-14
Feigenbaum number, 126
Ferrari, Ludovico, 216
First order ordinary differential equations: applications of, 47-64, 74-89 Bernoulli, 73 direction field, 3, 5 exact, 89-93 existence and uniqueness theorems, 65, 66, 106 proof of, 106-113 general solution of, 11, 30, 36,
70, 71
graphical construction of integral curves, 71 homogeneous, 24, 46-47 implicit solutions, 70-71 integrating factor for, 30-31, 35-36, 93-95, 96 interval of definition, 37, 43, 69-70 linear, 24, 29-39 nonlinear, 64-72 numerical solution of, see Numerical methods
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