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

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singular, 656-663 continuous spectrum, 660 Successive approximations, method of, 106, 401

Index

745

Superposition principle, 140,

344, 368, 600, 663 Sylvester, James, 348 Systems: of algebraic equations, 357-360, 542-543 of differential equations, 18 autonomous, 471-472 existence and uniqueness theorem, 343 initial conditions, 342 linear, 343 nonlinear, 343 numerical solution of, 455-458 reduction to, 341-342 solution of, 342 of linear first order equations, 368-418 definition of, 343 diagonalization, 398-400, 411-413 Euler, 382, 391, 408 existence and uniqueness theorem, 343 fundamental matrix, 393-401, 405-406 fundamental set of solutions, 370

general solution of, 370 homogeneous, 343,

368-372 homogeneous with constant coefficients, 373-410 complex eigenvalues, 384-393, 464-467 real and unequal eigenvalues, 373-383, 460-462

repeated eigenvalues, 401-410, 462-464 nonhomogeneous, 343, 411-418 superposition of solutions, 368

undetermined coefficients,

413-414 variation of parameters,

414-416

T

Tangent line method, see Euler method Tautochrone, 336-337 Taylor, Brook, 234 Taylor series, 154, 234

for functions of two variables,

434

Telegraph equation, 619, 627, 653 Thermal diffusivity, 574, 616 Threshold models, 81-84 Trajectories, 342, 373, 460, 467-468, 475-477 almost linear systems,

483-484 linear systems, 460-467 Transfer function, 335 Transient solution, 203, 583 Translation of function, 311 Transpose of matrix, 348 Triangular wave, 551-552, 553-555, 571 Truncation error, global, 424,

426, 445-447 local, 424

U

Undetermined coefficients, 171-178, 222-226 for systems of equations,

413-414 Uniqueness theorems, see

Existence and uniqueness theorems Unit step function, 310 Laplace transform of, 311

V

Van der Pol equation, 478, 526-530 Variation of parameters, 25, 39, 179-185, 226-230

for systems of equations,

414-416 Vector space, 151 Vectors, 151, 349 inner product, 351 length, 351

linear dependence and independence of,

360-362 magnitude, 351 multiplication of, 351 orthogonality, 351 Verhulst, Pierre F., 76 Verhulst equation, 76 Vibrations:

of elastic bar, 628-629 of elastic membrane, 604, 664-666 of elastic string, 591-604, 617-619 of spring-mass system,

186-195,200-207,306,

392

two degrees of freedom, 220,

345, 393 Volterra, Vito, 504 Volterra integral equation, 336

W

Wave equation, 17, 591, 617-619 in polar coordinates, 664, 667 solution of, 602 see also Elastic string, derivation of wave equation Wavelength, of vibrating string, 594

Wave velocity, 591, 601, 619 Wronski, Jozef, 141 Wronskian, 141, 211 Abel’s identity for, 149, 213 for systems of equations, 369 Abel’s identity, 370

Y

Yield, maximum sustainable, 87

Student Solutions Manual

Charles W. Haines

Rochester Institute of Technology

to accompany

Elementary Differential Equations, 7e and

Elementary Differential Equations and Boundary Value Problems, 7e

William E. Boyce Richard C. DiPrima

Rensselaer Polytechnic Institute

John Wiley & Sons, Inc.

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PREFACE

This supplement has been prepared for use in conjunction with the seventh editions of ELEMENTARY DIFFERNTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS and ELEMENTARY DIFFERENTIAL EQUATIONS, both by W.E. Boyce and R.C. DiPrima. The supplement contains a sampling of the problems from each section of the text. In most cases the complete details in determining the solutions are given while in the remainder of the problems helpful hints are provided. The problems chosen in each section represent, wherever possible, the variety of applications and types of examples that are covered in the written material of the text, thereby providing the student a complete set of examples from which to learn.

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