# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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functions, 291, 661, 662 of Chebyshev polynomials, 663

of eigenfunctions of Sturm-Liouville problems, 632, 660 of functions, 549 of Legendre polynomials, 255, 662, 676 of sine and cosine functions, 549-550

of vectors, 351 Orthonormal set, 633, 669 Overdamped motion, 193

P

Parseval equation, 564, 571, 604, 676

Partial differential equation, definition of, 17 See also Heat conduction equation, Laplace’s equation, Wave equation Partial fraction expansion, 302, 305,306,309 Particular solution, 170, 212 Pendulum equation: generalized nonlinear undamped, 519 linear undamped, 19 nonlinear damped, 473-475, 481-482, 483,484-487, 489,519-520 nonlinear undamped, 19, 478, 488-489, 490-491, 511-513,516 period, 490-491 Period, of nonlinear undamped pendulum, 490-491 of simple harmonic motion, 190

Periodic boundary conditions, 638, 674 Periodic forcing terms, 563 Periodic functions, 548 cosine and sine, 548-549 derivative of, 557 fundamental period, 548 integral of, 556-557 Laplace transform of, 315 linear combination of, 549 product of, 549 Periodic solutions of autonomous systems, 503-511, 521-531 Phase, of simple harmonic motion, 190 Phase plane, 373, 460 Phase plot, 207 Phase portrait, 373, 460

Picard, Charles Emile, 106 Picard, method of, 106 Piecewise continuous functions,

295, 558 Poincare, Henri, 459 Poincare-Bendixson theorem,

525

Population, growth of, 74-84 Positive definite function, 513 Positive semidefinite function,

513

Potential equation, 604-613 See also Laplace’s equation Power series, properties of, 232-235 Predator-prey equations, 18, 340-341,503-511 Predictor-corrector method, 441, 458

Proper node, 462

Q

Quasi frequency, 192 Quasi period, 193

R

Radioactive decay, 16, 59 Radiocarbon dating, 59 Radius of convergence, 232, 251 Ramp loading, 320 Rate function, 5 Rate of growth (decline), 5, 75 intrinsic, 76 Rayleigh equation, 531 Rectified sine wave, 316 Recurrence relation, 240, 268, 269, 274 Reduction of order, 165-166, 212 Reduction to systems of

equations, 341-342 Region of asymptotic stability, 487, 498,516-519 Regular singular point, 255-260 at infinity, 260 Resistance, electric, 195, 196 see also Damping force Resonance, 202-203, 323-324 Rodrigues’ formula, 255

744

Index

Round-off error, 424, 429, 445-447 Row reduction, 352 Runge, Carl D., 436 Runge-Kutta method, 436-439, 456

S

Saddle point, 375, 389, 461-462 Saturation level, 78 Sawtooth wave, 316, 567-568, 571

Scalar product, see Inner product Schaefer model for fish population, 86 Schrodinger, Erwin, 326 Self-adjoint: boundary value problem, 637-639, 660 equation, 147 matrix, 365 Separable equations, 24, 40-46 Separated boundary conditions, 630

Separation constant, 575, 585, 593, 606, 623, 665 Separation of variables, 573 further remarks, 664-669 for heat conduction equation,

575, 585 for Laplace’s equation, 606 in polar coordinates, 609 for wave equation, 593 in polar coordinates, 667 Separatrix, 487, 498 Series of eigenfunctions,

635-636, 642, 660-661, 672

Series solution: existence theorem for, 250, 277-278 first order equations, 254 indicial equation, 269, 273, 274, 278 near an ordinary point, 238-255 near a regular singular point, 267-280

recurrence relation, 240, 268, 269, 274 when roots of indicial equation: are equal, 270, 276-277, 278, 280-284 differ by integer, 270, 277, 278, 285-289 Shift of index of summation, 235-236 Similarity transformation, 397-398 Similar matrices, 398 Simple eigenvalues, 364 Simple harmonic motion, 190 Simply connected region, 91 Simpson’s rule, 437 Sine series, 567 Singular matrix, 352 Singular point, 238, 250,

255-260 irregular, 258, 279-280 regular, 255-260 at infinity, 260 Sink, nodal, 461 spiral, 465 Slope field, see Direction field Solution of ordinary differential equation, 20 general solution of linear

equations, 11, 30, 36, 70,

71, 133, 142, 143, 155, 163, 170,211 systems, 370 implicit, 70-71 of systems of equations, 342 Source, nodal, 461 spiral, 465 Spiral point, 386, 390, 464-466, 479-480 Spring-mass system, 186-195, 200-207, 306, 392-393 two degrees of freedom, 220, 345, 393 Square integrable function, 672 Square wave, 315, 316, 560-562, 571 Stability:

for almost linear systems, 483-484 asymptotic, 79, 117, 119, 468, 471,473,475,514-515, 524

see also Converging solutions basin of attraction, 487, 498, 516-519 of critical point, 468, 471, 472-473,475,479,514 definition of, 472 exchange of, 120 globally asymptotically stable, 487

Liapunov theorems, 514 for linear systems, 468, 479-480 of numerical method, 448-453 orbital, 524

region of asymptotic stability, 487, 498,516-519 semistable, 84, 524 stable, 468,471,472, 475,514, 524

unstable, 79, 117, 119, 468,

471,472, 475,514,524 see also Diverging solutions Stairstep diagram, 120 Star point, 462

Steady state solution, 203, 582 Step functions, 310 Stiff equations, 450-451 Stokes, George Gabriel, 62 Stokes’ law, 62 Strange attractor, 536 Sturm, Charles, 629 Sturm-Liouville boundary value problems, 629-641 eigenfunctions orthogonal, 632 eigenvalues, real, 631 simple, 633 nonhomogeneous, 641-645 self-adjointness of, 637-639, 660

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