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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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of sawtooth wave, 316 of square wave, 315, 316 of step function, 311 table of, 304
translation formula, 311, 313 Legendre, Adrien Marie, 149,
254
Legendre equation of order alpha, 147, 152, 238, 252,
254-255, 256, 258, 271, 309, 657, 662, 669 Legendre polynomials, 254-255, 662, 676 Leibniz, Gottfried Wilhelm, 24, 63, 73, 336, 571 Length, of a vector, 351 L'Hospital, Marquis de, 63 Liapunov, Aleksandr M., 511 Liapunov function, 514 Liapunov's second method, 511-521 Libby, Willard F., 59 Lienard equation, 491, 520 Limit cycle, 524 Linear dependence and independence, of functions, 147-153, 211 of vector functions, 362 of vectors, 360 Linear operator, 138, 226, 298, 303
Linear ordinary differential equations: adjoint equation, 146 change of independent
variable, 159-160, 264, 266, 291 characteristic equation, 132, 214, 303 complex roots, 153, 217 real and equal roots, 161,
218
real and unequal roots, 132, 215
complementary solution, 170 definition of, 19 Euler equation, 160, 169, 260-267 exact, 146
742
Index
existence and uniqueness theorems, 65, 138, 210, 250, 277-278, 343 first order, 24, 29-39 fundamental set of solutions, 142, 211, 370 general solution of, 11, 25, 30, 36, 70, 71, 133, 142, 143, 155, 163, 170, 211, 370 homogeneous equation with constant coefficients, 25, 129-137, 153-159,
160-169, 214-219 integrating factor, 25, 30-31, 35-36, 146 nonhomogeneous equation, 130, 169-185, 211-212, 222-230, 411-418 ordinary point, 238, 250, 254, 260
particular solution, 170, 212 reduction of order, 165-166, 212
self-adjoint, 147 series solution of, see Series solution singular point, 238, 250, 255-260 systems of, see Systems undetermined coefficients, 171-179, 222-226 variation of parameters, 25, 39, 179-185, 226-230 Liouville, Joseph, 106, 629 Logarithmic decrement, 198 Logistic equation, 76, 118, 511 Logistic growth, 75-81, 83-84 Lorenz, Edward N., 532 Lorenz equations, 532-539 Lotka, Alfred James, 504 Lotka-Volterra equations, 18, 340-341, 503-511
M
Magnitude, of a vector, 351 Malthus, Thomas, 75 Mathematical model, 2, 47-49 analysis of, 48-49
comparison with experiment, 49
construction of, 7, 14, 48 Matrices, 348-367 addition of, 349 adjoint, 348 augmented, 358 conjugate, 348 diagonalizable, 397-398 eigenvalues of, 362-366, 544 eigenvectors of, 362-366, 544 equality of, 349 exponential, 396-397, 400 fundamental, 393-401, 405-406 Gaussian elimination, 352 Hermitian, 248, 365-366, 379, 397, 402, 637, 653 identity, 351 inverse, 352-354 invertible, 352
Jordan form of, 406-407, 409, 410
multiplication of, 349-350 multiplication by numbers, 349 noninvertible, 352 nonsingular, 352 row reduction of, 352 self-adjoint, 365 similar, 398 singular, 352 subtraction of, 349 transpose, 348 zero, 349 Matrix functions, 354-355 Mean convergence, 672 Mean square error, 671 Millikan, Robert A., 62 Mixing problems, 49-51, 54-55 Mode, natural (of vibrating string), 594 Modified Euler formula, 435 Moulton, Forest Ray, 440 Multiplicity of eigenvalue, 364 Multistep method, 439-445
N
Negative definite function, 513
Negative semidefinite function, 513
Neumann, Karl Gottfried, 605 Neumann problem, 605 for circle, 613 for rectangle, 612 Newton, Isaac, 24, 63 Newton’s law: of cooling, 60 of motion, 2, 187, 345, 618 Node, 378, 390, 461 See also Improper node;
Proper node Nonhomogeneous algebraic equations, 358 Nonhomogeneous boundary
value problems, 542-543,
641-656 Fredholm alternative, 644 solution, by eigenfunction expansion, 641-653 by Green’s function, 653-656 Nonhomogeneous linear
differential equations,
130, 169-185, 211-212, 222-230, 334-335, 343, 411-418 Noninvertible matrix, 352 Nonlinear ordinary differential equations: autonomous systems, 471-539 definition of, 19, 130 existence and uniqueness theorems, 66, 106, 343 first order, 64-72 methods of solving, 40-47, 89-96, 137 periodic solutions of, 503-511, 521-531 /= f (t, y), 137 / = f (y, y), 137 Nonsingular matrix, 352 Normalization condition, 633 Nullcline, 499 Numerical dependence, 452 Numerical methods, 419-458 Adams-Bashforth formula,
440
Index
743
Adams-Moulton formula, 441 adaptive, 427, 433 backward differentiation formulas, 443-444 backward Euler formula, 422-424 comparison of, 444 convergence of, 424 effect of step size, 426, 445-447 error see Error Euler, 96-105,419-429, 455-457 Heun, 431
improved Euler, 430-435 modified Euler, 435 multistep, 439-445 one-step, 439
predictor-corrector, 441, 458 Runge-Kutta, 435-439, 456 stability of, 448-453 for stiff equations, 450-451 for systems of first order equations, 455-458 vertical asymptotes, 447-448 Numerical stability, 448-453
O
Odd function, 565 Odd periodic extension, 568 One-step method, 439 Orbital stability, 524 Order of differential equation, 18 Ordinary differential equation, definition of, 17 Ordinary point, 238, 250, 254 at infinity, 260 Orthogonality, of Bessel
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
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