Edwards differential equations and linear algebra 4th ed 2018

DIFFERENTIAL EQUATIONS & LINEAR ALGEBRA Fourth Edition C Henry Edwards David E Penney The University of Georgia David T Calvis Baldwin Wallace University Director, Courseware Portfolio Management: Deirdre Lynch Executive Editor: Jeff Weidenaar Editorial Assistant: Jennifer Snyder Managing Producer: Karen Wernholm Content Producer: Tamela Ambush Producer: Nicholas Sweeny Field Marketing Manager: Evan St Cyr Product Marketing Manager: Yvonne Vannatta Marketing Assistant: Jon Bryant Senior Author Support/Technology Specialist: Joe Vetere Manager, Rights Management: Gina M Cheselka Manufacturing Buyer: Carol Melville, LSC Communications Composition: Dennis Kletzing, Kletzing Typesetting Corp Cover Design: Studio Montage Cover Image: Thegoodly/Getty Images Photo Credits: Icon/Logo, Rashadashurov/Fotolia; Screenshots from Texas Instruments Incorporated Courtesy of Texas Instruments Incorporated; p 45, Screenshot by Wolfram Alpha LLC c 2018, 2010, 2005 by Pearson Education, Inc or its affiliates All Rights Reserved Copyright Printed in the United States of America This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise For information regarding permissions, request forms and the appropriate contacts within the Pearson Rights & Permissions Department, please visit www.pearsoned.com/permissions/ PEARSON, ALWAYS LEARNING is an exclusive trademark owned by Pearson Education, Inc or its affiliates in the United States and/or other countries Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc or its affiliates, authors, licensees or distributors Library of Congress Cataloging-in-Publication Data Names: Edwards, C Henry (Charles Henry), 1937– j Penney, David E j Calvis, David Title: Differential equations & linear algebra / C Henry Edwards, David E Penney, The University of Georgia; with the assistance of David Calvis, Baldwin-Wallace College Description: Fourth edition j Boston : Pearson, [2018] j Includes bibliographical references and index Identifies: LCCN 2016030491 j ISBN 9780134497181 (hardcover) j ISBN 013449718X (hardcover) Subjects: LCSH: Differential equations j Algebras, Linear Classification: LCC QA372 E34 2018 j DDC 515/.35 dc23 LC record available at https://lccn.loc.gov/2016030491 16 ISBN 13: 978-0-13-449718-1 ISBN 10: 0-13-449718-X CONTENTS Application Modules Preface ix CHAPTER First-Order Differential Equations 1.1 1.2 1.3 1.4 1.5 1.6 CHAPTER 2.1 2.2 2.3 2.5 2.6 Differential Equations and Mathematical Models Integrals as General and Particular Solutions 10 Slope Fields and Solution Curves 17 Separable Equations and Applications 30 Linear First-Order Equations 46 Substitution Methods and Exact Equations 58 Mathematical Models and Numerical Methods 2.4 CHAPTER vi Population Models 75 Equilibrium Solutions and Stability 87 Acceleration-Velocity Models 94 Numerical Approximation: Euler’s Method A Closer Look at the Euler Method 117 The Runge–Kutta Method 127 Linear Systems and Matrices 3.1 3.2 3.3 3.4 3.5 3.6 3.7 75 106 138 Introduction to Linear Systems 147 Matrices and Gaussian Elimination 146 Reduced Row-Echelon Matrices 156 Matrix Operations 164 Inverses of Matrices 175 Determinants 188 Linear Equations and Curve Fitting 203 iii iv Contents CHAPTER Vector Spaces 4.1 4.2 4.3 4.4 4.5 4.6 4.7 CHAPTER 5.1 5.2 5.3 5.5 5.6 CHAPTER The Vector Space R3 211 The Vector Space Rn and Subspaces 221 Linear Combinations and Independence of Vectors Bases and Dimension for Vector Spaces 235 Row and Column Spaces 242 Orthogonal Vectors in Rn 250 General Vector Spaces 257 Higher-Order Linear Differential Equations 5.4 CHAPTER 211 6.2 6.3 339 Introduction to Eigenvalues 339 Diagonalization of Matrices 347 Applications Involving Powers of Matrices 354 Linear Systems of Differential Equations 7.1 7.2 7.3 7.4 7.5 7.6 7.7 265 Introduction: Second-Order Linear Equations 265 General Solutions of Linear Equations 279 Homogeneous Equations with Constant Coefficients 291 Mechanical Vibrations 302 Nonhomogeneous Equations and Undetermined Coefficients 314 Forced Oscillations and Resonance 327 Eigenvalues and Eigenvectors 6.1 228 365 First-Order Systems and Applications 365 Matrices and Linear Systems 375 The Eigenvalue Method for Linear Systems 385 A Gallery of Solution Curves of Linear Systems 398 Second-Order Systems and Mechanical Applications 424 Multiple Eigenvalue Solutions 437 Numerical Methods for Systems 454 Contents CHAPTER CHAPTER Matrix Exponential Methods 8.1 8.2 8.3 10 9.1 9.2 9.3 10.1 10.2 10.3 10.5 11 11.2 11.3 11.4 526 557 Laplace Transforms and Inverse Transforms 557 Transformation of Initial Value Problems 567 Translation and Partial Fractions 578 Derivatives, Integrals, and Products of Transforms 587 Periodic and Piecewise Continuous Input Functions 594 Power Series Methods 11.1 503 Stability and the Phase Plane 503 Linear and Almost Linear Systems 514 Ecological Models: Predators and Competitors Nonlinear Mechanical Systems 539 Laplace Transform Methods 10.4 CHAPTER Matrix Exponentials and Linear Systems 469 Nonhomogeneous Linear Systems 482 Spectral Decomposition Methods 490 Nonlinear Systems and Phenomena 9.4 CHAPTER 469 604 Introduction and Review of Power Series 604 Power Series Solutions 616 Frobenius Series Solutions 627 Bessel Functions 642 References for Further Study 652 Appendix A: Existence and Uniqueness of Solutions 654 Appendix B: Theory of Determinants 668 Answers to Selected Problems 677 Index 733 v APPLICATION MODULES The modules listed below follow the indicated sections in the text Most provide computing projects that illustrate the corresponding text sections Many of these modules are enhanced by the supplementary material found at the new Expanded Applications website, which can be accessed by visiting goo.gl/BXB9k4 For more information about the Expanded Applications, please review the Principal Features of this Revision section of the preface Computer-Generated Slope Fields and Solution Curves 1.4 The Logistic Equation 1.5 Indoor Temperature Oscillations 1.6 Computer Algebra Solutions 1.3 2.1 2.3 2.4 2.5 2.6 Logistic Modeling of Population Data Rocket Propulsion Implementing Euler’s Method Improved Euler Implementation Runge-Kutta Implementation Automated Row Operations 3.3 Automated Row Reduction 3.5 Automated Solution of Linear Systems 3.2 Plotting Second-Order Solution Families Plotting Third-Order Solution Families 5.3 Approximate Solutions of Linear Equations 5.5 Automated Variation of Parameters 5.6 Forced Vibrations 5.1 5.2 7.1 7.3 7.4 7.5 7.6 7.7 8.1 8.2 Gravitation and Kepler’s Laws of Planetary Motion Automatic Calculation of Eigenvalues and Eigenvectors Dynamic Phase Plane Graphics Earthquake-Induced Vibrations of Multistory Buildings Defective Eigenvalues and Generalized Eigenvectors Comets and Spacecraft Automated Matrix Exponential Solutions Automated Variation of Parameters Phase Plane Portraits and First-Order Equations Phase Plane Portraits of Almost Linear Systems 9.3 Your Own Wildlife Conservation Preserve 9.4 The Rayleigh, van der Pol, and FitzHugh-Nagumo Equations 9.1 9.2 vi Application Modules Computer Algebra Transforms and Inverse Transforms 10.2 Transforms of Initial Value Problems 10.3 Damping and Resonance Investigations 10.5 Engineering Functions 10.1 11.2 11.3 Automatic Computation of Series Coefficients Automating the Frobenius Series Method vii This page intentionally left blank PREFACE T he evolution of the present text is based on experience teaching introductory differential equations and linear algebra with an emphasis on conceptual ideas and the use of applications and projects to involve students in active problem-solving experiences Technical computing environments like Maple, Mathematica, MATLAB , and Python are widely available and are now used extensively by practicing engineers and scientists This change in professional practice motivates a shift from the traditional concentration on manual symbolic methods to coverage also of qualitative and computer-based methods that employ numerical computation and graphical visualization to develop greater conceptual understanding A bonus of this more comprehensive approach is accessibility to a wider range of more realistic applications of differential equations Both the conceptual and the computational aspects of such a course depend heavily on the perspective and techniques of linear algebra Consequently, the study of differential equations and linear algebra in tandem reinforces the learning of both subjects In this book we therefore have combined core topics in elementary differential equations with those concepts and methods of elementary linear algebra that are needed for a contemporary introduction to differential equations Principal Features of This Revision This 4th edition is the most comprehensive and wide-ranging revision in the history of this text We have enhanced the exposition, as well as added graphics, in numerous sections throughout the book We have also inserted new applications, including biological Moreover we have exploited throughout the new interactive computer technology that is now available to students on devices ranging from desktop and laptop computers to smartphones and graphing calculators While the text continues to use standard computer algebra systems such as Mathematica, Maple, and MATLAB, we have now added the Wolfram j Alpha website In addition, this is the first edition of this book to feature Python, a computer platform that is freely available on the internet and which is gaining in popularity as an all-purpose scientific computing environment However, with a single exception of a new section inserted in Chapter (noted below), the class-tested table of contents of the book remains unchanged Therefore, instructors notes and syllabi will not require revision to continue teaching with this new edition A conspicuous feature of this edition is the insertion of about 80 new computergenerated figures, many of them illustrating interactive computer applications with slider bars or touchpad controls that can be used to change initial values or parameters in a differential equation, and immediately see in real time the resulting changes in the structure of its solutions ix 730 Answers to Selected Problems X 20 y1 x/ D x 1=3 nD0 y2 x/ D X 1/n 2n x n , nŠ




3n C 1/ y 21 y1 x/ D x C X nD1 1=2 X 1C x 2n nŠ




4n 3/ ! y2 ! 1/n x 2n , nŠ

13


4n C 5/ nD1 ! X 1/n x 2n 1C nŠ




4n 1/ 22 y1 x/ D x 3=2 C y1 ! x 2n , nŠ

11


4n C 3/ nD1 y2 x/ D x 1 cosh 2x , y2 x/ D sinh 2x x x 1/n 2n x n nŠ




3n 1/ nD0 y2 x/ D x 28 y1 x/ D X 29 y1 x/ D x x x cos , y2 x/ D sin x x y nD1 ! x 2n , 23 y1 x/ D x 1C 2n
nŠ
19
31


12n C 7/ nD1 ! X x 2n y2 x/ D x 2=3 C n
nŠ

17


12n 7/ y1 0.5 X 1=2 y2 2π 4π x –0.5 nD1 24 y1 x/ D x 1=3 C X nD1 X y2 x/ D C nD1 1/n x 2n 2n
nŠ

11


6n nD0 X nD1 X 26 y1 x/ D x 1=2 X nD1 27 y1 x/ D y y2 1/ π x=2 , x π y1 –1 1/n x n 2n nD0 y2 x/ D C 1/n x 2n ,
nŠ

13


6n C 1/ X 1/n x n D x 1=2 e nŠ
2n 25 y1 x/ D x 1=2 y2 x/ D C 2n 30 y1 x/ D cos x , y2 x/ D sin x ! 1/ŠŠ 31 y1 x/ D x 1=2 cosh x , y2 x/ D x 1=2 sinh x y   x 2n D x 1=2 exp 12 x , nŠ
2n 2n x 2n



4n y1 1/ y2 1 cos 3x , y2 x/ D sin 3x x x y 32 y1 x/ D x C y2 x/ D x 2π –1 y 4π x 33 y1 x/ D x x2 , ! 5x C


48 ! 10x C 10x C 5x C C


, 1=2 y2 x y2 x/ D x 1=2 C 5x 11x 20 15x 11x 671x C C


224 24192 ! 731 Answers to Selected Problems ! x2 x4 C C


, 42 1320 34 y1 x/ D x y2 x/ D x 1=2 Appendix A 7x 19x C C


24 3200 y0 D 3, y1 D C 3x , y2 D C 3x C 32 x , y3 D C 3x C 23 x C 12 x , y4 D C 3x C 32 x C 12 x C 18 x ; y.x/ D 3e x ! Section 11.4 J4 x/ D x x 10 24/J0 x/ C x /J1 x/ x Z 11 x J1 x/ C xJ0 x/ J0 x/ dx C C 12 .x 9x /J1 x/ C 3x 9x/J0 x/ C Z xJ1 x/ C J0 x/ dx C C 13 .x 14 4x/J1 x/ C 2x J0 x/ C C 15 2xJ1 x/ 17 18 J0 x/ dx C C x J0 x/ C C 16 3x J1 x/ C 3x 4x Z x /J0 x/ x/ C 8x 16x/J Z 2J1 x/ C J0 x/ dx C C Z J0 x/ dx C C x /J0 x/ CC 19 y.x/ D x Œc1 J0 x/ C C2 Y0 x/ 20 y.x/ D Œc1 J1 x/ C c2 Y1 x/ x  21 y.x/ D x c1 J1=2 3x / C c2 J 1=2 3x /   1=2 1=2 22 y.x/ D x c1 J2 2x / C c2 Y2 2x /

  23 y.x/ D x 1=3 c1 J1=3 13 x 3=2 C c2 J 1=3 13 x 3=2   24 y.x/ D x 1=4 c1 J0 2x 3=2 / C c2 Y0 2x 3=2 / 25 y.x/ D x 1 Œc1 J0 x/ C c2 Y0 x/  26 y.x/ D x c1 J1 4x 1=2 / C c2 Y1 4x 1=2 /   1=2 3=2 27 y.x/ D x c1 J1=2 2x / C c2 J 1=2 2x 3=2 / 

 28 y.x/ D x 1=4 c1 J3=2 25 x 5=2 C c2 J 3=2 25 x 5=2 

 29 y.x/ D x 1=2 c1 J1=6 13 x C c2 J 1=6 13 x 

 30 y.x/ D x 1=2 c1 J1=5 45 x 5=2 C c2 J 1=5 45 x 5=2 y0 D 1, y1 D x , y2 D x C 12 x , y3 D x C 12 x 16 x , y4 D x C 12 x 16 x C 24 x ; y.x/ D exp y0 D 0, y1 D 2x , y2 D 2x C 2x , y3 D 2x C 2x C 34 x , y4 D 2x C 2x C 43 x C 23 x ; y.x/ D e 2x x2, x2 y0 D 0, y1 D y2 D y4 D x C 21 x C 16 x C x2
C 1 6x , C 12 x , y3 D x C 12 x 24 x ; y.x/ D exp.x / y0 D 1, y1 D C x/ C 21 x , y2 D C x C x / C 61 x ,
y3 D C x C x C 13 x C 24 x ; x y.x/ D 2e x D C x C x C 13 x C


11 y0 D 1, y1 D C x , y2 D C x C x / C 13 x , y3 D C x C x C x / C 23 x C 13 x C 19 x C 63 x ; D C x C x C x C x C x C


y.x/ D x x , 12 y0 D 1, y1 D C 12 x , y2 D C 12 x C 83 x C 18 x C 64 C


; y3 D C 12 x C 38 x C 16 x C 13 x 64 y.x/ D x/ 1=2         x0 x1 C 3t 13 D , D , y0 y1 C 3t     x2 C 3t C t D , y2 C 5t 12 t     x3 C 3t C 12 t C 13 t D y3 C 5t 12 t C 56 t  t  e C t et 14 x.t / D 16 y3 1/  0:350185 t e This page intentionally left blank INDEX Boldface page numbers indicate where terms are defined A Abel’s formula, 277, 289 Acceleration, 11 constant, 12 Addition of matrices, 164 associative law, 171 commutative law, 171 distributive law, 171 Addition of vectors, 212, 222 parallelogram law, 212, 223 triangle law, 212 Adjoint matrix, 199, 675 Air resistance, 20, 94 proportional to square of velocity, 97 proportional to velocity, 20, 95, 460 Airy equation, 625, 650 Airy function, 625 Algebra of inverse matrices, 178 Algebraic multiplicity, 354 Alligator population, 82 Almost linear system, 516 stability, 519 Amplification factor, 330 Amplitude, 305 Analytic function, 606 Apollo (satellite orbit), 462 Approximate logistic solution, 112 Argument (of complex number), 298 Arnold, David, 28 Artin, Emil, (1898–1962), 644 Associated eigenvector, 339 Associated homogeneous equation, 266, 279, 377 Asymptotic stability, 508 Augmented coefficient matrix, 148 Automobile: simplified model, 312 two-axle, 434 vibrations, 331, 336 Autonomous differential equation, 88 critical point, 88 equilibrium solution, 88 stable critical point, 89 unstable critical point, 89 Autonomous system, 503 linearized, 516 Auxiliary equation, see Characteristic equation Average error, 80 B Back substitution, 140, 146 algorithm, 151 Basic unit vector, 218 Basis, 235 as maximal linearly independent set, 236, 241 as minimal spanning set, 242 as uniquely spanning set, 242 for R3 , 218 for column space, 246 for row space, 244 for solution space, 240 orthogonal, 253 Batted baseball, 458 Beats, 329 Bernoulli equation, 61 Bessel, Friedrick W (1784–1846), 642 Bessel equation, 590, 604, 635, 639, 642 Bessel function: identities, 647 order 1, 639 order 12 , 290, 645 12 order 32 , 650 32 order n, integral order, second kind, 646 order p , 642 order p , first kind, 645 order zero, first kind, 636 order zero, second kind, 646 solutions in, 649 Bifurcation diagram, 92 Bifurcation point, 92, 523 Binomial series, 605, 615 Birkhoff, Garrett (1911–1996), 118 Birth rate, 75 Bounded population growth, 76 Brachistochrone problem, 44 Brine tank examples, 366, 389, 393, 395, 496 Broughton Bridge, 330 Buoy problem, 311 Bus orbit, 462 C Carbon-14, 36 Carrying capacity, 21, 78 Cart with flywheel, 327 Cascade of brine tanks, 54, 395, 483 Catenary problem, 44 Cauchy-Schwarz inequality, 251 Cayley-Hamilton theorem, 174, 361, 450, 477, 491, 498 Center: critical point, 412 of power series, 606 stable, 508 Central conic, 208 Chain (of generalized eigenvectors), 442, 444 Characteristic equation, 274, 292 of matrix, 340, 386 Characteristic value, see Eigenvalue Churchill, R V (1899–1987), 565, 592 Circular frequency, 305, 308 Clairaut equation, 71 Clarinet reed, 551 Clarke, Arthur (1917–2008), 17 Closed trajectory, 509 Clepsydra, 43 Coddington, E A (1920–1991), 633 Coefficient matrix, 147 augmented, 148 of a first-order system, 376 Coexistence of species, 532 733 734 Index Cofactor, 190 expansion, 190 matrix, 199, 675 Column (of a matrix), 147 Column rank, 244 and row rank, 247 Column space, 244 basis for, 246 Column vector, 148, 165, 169 Comet orbit, 467 Compartmental analysis, 388 Competing species, 529, 534 Competition situation, 80 Competition system, 529, 535 Complementary function, 287 Complete eigenvalue, 437 Complete independence of eigenvectors, 353 Complete set of eigenvectors, 437 Complex eigenvalue, 391, 410 with negative real part, 414 with positive real part, 415 Complex-valued function, real and imaginary parts, 296 Componentwise addition, 212, 222 multiplication of vector by scalar, 213, 223 Compound interest, 36 Computational efficiency, 200 Conservation of energy, 303, 331 Constant acceleration, 12 Continuous dependence of solutions, 665 Convergence: of improper integral, 558 of power series, 605 Convolution (of functions), 587 Convolution property (of transforms), 588 Cooperation of species, 534 Coordinate plane, 214 space, 211 Corresponding eigenvalue, 339 Cramer’s rule, 188, 197, 674 Criterion for diagonalizability, 349 Criterion for exactness, 66 Critical damping, 307, 308 Critical point (of system), 503 asymptotic stability, 508 center, 508 classification, 520 isolated, 514 node, 506 saddle point, 400, 507 spiral point, 509 stability, 507 Crossbow, 94, 98, 125, 134 Cumulative error, 110 Curvature, 71 Curve fitting, 203 D Damped nonlinear vibrations, 542 Damped pendulum oscillations, 548 Damping constant, 302 Death rate, 75 Decay constant, 36 Defect (of eigenvalue), 439 Defective eigenvalue, 439 Dependence (linear), 214, 232, 270, 282, 378 Dependence on parameters, 91 Dependent variable missing, 68 Derivative (of matrix function), 375 Determinant, 188, 189, 190  2, 188  3, 189 n  n, 190 and invertibility, 196, 672 and matrix product, 196, 672 and elementary row operations, 192, 669 by cofactor expansion, 190 by elimination method, 194 by permutations, 669 coefficient, 188 row and column properties, 192 Vandermonde, 204, 289 Diagonal (principal), 162 Diagonal matrix, 174 Diagonalizable matrix, 349, 352 Diagonalization, criterion for, 349 Differential equation, autonomous, 88 Bernoulli, 61 Clairaut, 71 differential form, 65 Euler, 277 exact, 65 first-order, general solution, 10, 11, 35, 271, 272, 286, 379, 382, 386 homogeneous, 59, 279 implicit solution, 35 linear, 47, 50, 265, 279 logistic, 21, 44, 76, 80 matrix, 469 normal form, order, order n, ordinary, partial, particular solution, 10 Riccati, 70, 651 second-order, 11, 68 separable, 32 singular solution, 35 solution, Differential equations and determinism, 313 Differential form, 65 Differential operator, 293 Differentiation of transforms, 589 Dimension and rank, 248 Dimension of a vector space, 237 Direction field, 18, 505 Direction of flow, 413 Displacement vector, 425 Distance between points, 252 Distinct real eigenvalues, 387, 399, 402 Dog problem, 71 Doomsday versus extinction, 81, 536 Dot product, 250 Drag coefficient, 96 Drug elimination, 37 Duplication (undetermined coefficients), 319 Dynamic damper, 434 Dynamic phase plane graphics, 421 E Earth-Moon satellite orbits, 461 Earthquake vibrations, 336, 435 Echelon form, 150 Echelon matrix, 150 reduced, 157 Eigenspace, 345 Eigenvalue, 339 complete, 437 complex, 391, 410, 414 defective, 439 distinct real, 351, 387, 399, 402 for matrix, 339, 386 geometric significance of, 412 imaginary, 411 multiplicity 2, 441 multiplicity k , 437 repeated, 399 zero, 403, 409 Eigenvalue matrix, 348 Eigenvalue solution, 386 Eigenvalue-eigenvector algorithm, 341 Eigenvector, 339, 386 associated with distinct eigenvalues, 351 chain of, 442 complex conjugate, 410 linearly independent, 386 rank r generalized, 442 Eigenvector equation, 386 Index Eigenvector matrix, 348 Element (of a matrix), 147 Elementary matrix, 180 and row operations, 180 Elementary (row) operation, 142, 149 effect on determinant, 193, 669 Elimination constant, 37 Elimination method: for linear systems, 140, 148 Gauss-Jordan, 157 Gaussian, 152 Elliptic integral, 548 Elliptical orbit, 412 Engineering functions, 603 Entry (of a matrix), 147 Equilibrium position, 302 Equilibrium solution, 88 of system, 504 Equivalent systems and matrices, 150 Error: in Euler’s method, 117 in improved Euler method, 120 in Runge-Kutta method, 128 propagation, 112 Error function, 53 Escape velocity, 100 Euclidean space, inner product, 251 Euler, Leonhard (1707–1783), Euler (differential) equation, 277 Euler’s formula, 296 Euler’s method, 107 algorithm, 107 cumulative error, 110 error in, 117 for systems, 454 improved, 119 local error, 110 roundoff error, 111 Euler’s theorem, 565 Exact equation, 65 Exactness, criterion for, 66 Existence, uniqueness of solutions: first-order equation, 23 linear system, 371, 662, 663 nth-order equation, 280 second-order equation, 268 Explosion-extinction equation, 81, 89 Exponential growth, see Natural growth Exponential matrix, 472 computation of, 478 Exponential order, 563 Exponential series, 606 Exponents (at a regular singular point), 632 External force, 303 periodic, 431 vector, 431 Extinction situation, 82, 359 F Famous numbers, 117, 126, 136 Fibonacci sequence, 363, 615 Finite-dimensional vector space, 237 First-order equation, First-order system, 366, 370, 376 FitzHugh, Richard (1922–2007), 555 FitzHugh-Nagumo equations, 555 Flight trajectories, 63 Flywheel on cart, 327 Folia of Descartes, 524 Forced motion, 303 Forced oscillations: and resonance, 431 damped, 332 undamped, 327 Forced vibrations, 266 Formal multiplication of series, 606 Fox-rabbit example, 357 Fps units, 13 Free motion, 303 damped, 303, 307 undamped, 303, 304 Free oscillations, 428 Free variables, 151, 239 Free vibrations, 266 Frequency, 305 resonance, 432 Frobenius, Georg (1848–1919), 631 Frobenius series, 631 Frobenius series solutions, 633 From the Earth to the Moon, 100 Function space, 259 Fundamental matrix, 469 Fundamental theorem of algebra, 292, 341 Funnel, 89 of homogeneous system, 379, 386 of nonhomogeneous equation, 287 of nonhomogeneous system, 382, 482 Generalized eigenvector, 442 Geometric multiplicity, 354 Geometric series, 605 Geometric significance of eigenvector, 412 Global existence of solutions, 659 Graphical method, 18 H Hailstone problem, 55 Half-life, 38 Halley’s comet, 375, 467 Hard spring, 540 Harvesting a logistic population, 90 Harvesting and restocking, 127 Heaviside, Oliver (1850–1925), 565 Hermite equation, 624 Hermite polynomial, 624 Higher-order systems, 458 Hodgkin, A L (1914–1998), 554 Hodgkin-Huxley model, 555 Hole-through-Earth problem, 311 Homicide victim problem, 43 Homogeneous: first-order equation, 59 linear system of first-order equations, 370, 385 nth-order equation, 279, 291 second-order equation, 266 second-order system, 427, 496 Homogeneous linear (algebraic) system, 160 uniqueness of solution, 162 Hooke’s law, 302, 425, 507, 539 Hopf bifurcation, 524 Huxley, A F (1917–2012), 554 Hypergeometric equation, series, 640 G g (earth’s surface gravitational acceleration), 13 G (universal gravitational constant), 99 Gallery of phase plane portraits, 417–418 Galvani, Luigi (1737–1798), 554 Gamma function, 559, 643 Gauss, hypergeometric equation, 640 Gaussian elimination algorithm, 152 Gauss-Jordan elimination algorithm, 157 General population equation, 76 General solution, 10, 11, 35, 271 of homogeneous equation, 272, 286 735 I Identity matrix, 162, 175 Identity principle: for polynomials, 260 for power series, 609 Imaginary eigenvalues, 411 Implicit solution, 34 Improper integral, 401, 558 Improper node, 401, 506 Improved Euler method, 119, 120 algorithm, 119 error in, 120 for systems, 455 736 Index Independence: see Linear independence Independent variable missing, 69 Indicial equation, 632 Indoor temperature oscillations, 56 Infinite-dimensional vector space, 237 Initial condition, 3, Initial position, 12 Initial value problem, 7, 268, 280, 381 and elementary row operations, 380 Laplace transform solution, 568 order n, 280 Initial vector, 356 Initial velocity, 12 Inner product, 250 Integrating factor, 46 Integration of transforms, 591 Interpolating polynomial, 204 Intersection of subspaces, 228 Inverse Laplace transform, 561 uniqueness of, 565 Inverse matrix, 177 adjoint formula, 199, 676 algebra, 178 algorithm, 181 solution of linear system, 179 uniqueness, 177 Inverse of  matrix, 178 Inverse-square law of gravitation, 99, 373 Invertible matrix, 177 and determinants, 196 and row operations, 181 Irregular singular point, 629 Isolated critical point, 514 J Jacobian matrix, 516 Joint proportion situation, 80 Jordan block (matrix), 449 Jordan normal form, 449 Jump, 562 K Kansas City, 330 Kepler, Johannes (1571–1630), 373 laws of planetary motion, 373, 466 Kinematic formula, 17 Kinetic energy, 303, 331 Kronecker delta, 492 Kutta, Wilhelm (1867–1944), 127 L Lakes Erie, Huron, Ontario, 52 Laplace, Pierre Simon de (1749–1827), 565 Laplace transform, 558 and convolution, 588 and initial value problems, 568 and linear systems, 571 differentiation of, 589 existence, 563 integration of transforms, 591 inverse, 561 inverse transforms of series, 594 linearity, 560 notation, 561 of derivative, 568 of higher derivatives, 569 of integral, 574 of periodic function, 597 of square and triangular wave functions, 598 products of transforms, 587 translation on the s -axis, 579 translation on the t -axis, 594 uniqueness of inverse, 565 Law of cosines, 202 Leading entry, 150, 239 Leading variables, 151, 239 Legendre polynomial, 623 Legendre’s equation, 290, 604, 622 Length of vector, 213 Leonardo Fibonacci (1175–1250?), 363 Limit cycle, 524 Limited environment situation, 80 Limiting population, 21, 78, 90, 358 Limiting velocity, 20 Linear approximation formula, 109 Linear combination, 167, 228 Linear differential equation: first-order, 47, 50 nth-order, 279 second-order, 265 Linear factor partial fractions, 579 Linear independence, 214, 216, 231, 270, 282 and orthogonality, 253 and unique linear combinations, 232 of vector-valued functions, 378 Linear independence and the determinant, 233, 234 Linear system (algebraic), 139 consistent, 139 equivalent, 150 homogeneous, 160 inconsistent, 139 nonhomogeneous, 161 number of solutions (three possibilities), 160 reduced echelon, 157 solution of, 139 upper triangular form, 143 Linear system (differential equations): associated homogeneous system, 377 eigenspace of, 345 eigenvalue method, 386 first-order, 370, 376 general solution, 379, 386 homogeneous, 370 nonhomogeneous, 370, 382, 482 second-order, 426, 496 stability, 518 Linearity of Laplace transform, 560 Linearized system, 516 Lipschitz continuous, 658 Local error, 110 Local existence of solutions, 663 Logarithmic decrement, 313 Logistic equation, 21, 44, 76 competition situation, 80 joint proportion situation, 80 limited environment situation, 80 with harvesting, 90 Logistic populations, interactions of, 534 Logistic prey population, 536 Lower triangular matrix, 194 Lunar lander, 12, 99, 457 M Maclaurin series, 606 Manchester (England), 330 Mass matrix, 425, 446 Mass-spring system, 266, 302, 309, 366, 425, 427, 432, 496, 507, 539, 570, 571, 580, 596 Mathematical model, modeling, Matrix, 147 addition, 164 adjoint, 199, 675 augmented, 148 coefficient, 147 cofactor, 190, 675 column, 147 determinant, 188 diagonal, 174 diagonalizable, 349, 352 echelon, 150 eigenvalue, 330, 386 eigenvector, 339, 386 element, 147 elementary, 180 elementary row operations, 149 entry, 147 equality, 164 exponential, 472 fundamental, 469 Index identity, 162, 172, 175 inverse, 177, 676 invertible, 177 Jacobian, 516 minor, 190 multiplication by scalar, 165 negative of, 165 nilpotent, 474 nonsingular, 183 norm, 658 orthogonal, 202 principal diagonal, 162 product, 168 projection matrices of, 491 reduced echelon, 157 row, 147 shape, 147 similar, 349 singular, 183 size, 147 spectral decomposition, 493 square, 161 stochastic, 357 trace, 347 transition, 356 triangular, 194 transpose, 195 with distinct eigenvalues, 352 zero, 172 Matrix algebra, 171 Matrix differential equation, 376, 469 Matrix equations, 182 Matrix exponential, 494, 499 Matrix exponential solution, 475 Matrix-valued function, 375 continuous, 375 differentiable, 375 derivative of, 375 Mechanical systems (modeling), 331 Method of elimination, 140 for determinants, 194 Method of Frobenius, 631 Method of successive approximations, 654 Method of undetermined coefficients, 314, 316, 320 for nonhomogeneous systems, 483 Mexico City, 330 Minimal spanning set, 242 Minor (of a matrix), 190 Mixture problems, 51 Mks units, 13 Modulus (of complex number), 298 Multiplication by an elementary matrix, 671 Multiplication of matrices, 168 and determinants, 196 associative law, 171 by scalars, 165 Multiplication of vector by scalar, 213 Multiplicity of eigenvalue, 354, 437 Multistory building, 435 Mutual extinction, 359 N n-space, 222 n-tuple, 222 n-vector, 166, 222 Natural frequency, 328, 428 Natural growth and decay, 35 Natural growth equation, 37 Natural mode of oscillation, 428 Newton, Sir Isaac (1642–1727), 95, 373 Newton’s law of cooling, 2, 38 Newton’s law of gravitation, 99, 373 Newton’s method, 97, 292 Newton’s second law of motion, 11, 15, 94, 266, 302, 327, 365, 425 Nilpotent matrix, 474, 498 Nodal sink (or source), 401, 402, 406, , 408, 506 Node (improper or proper), 401, 506 Nonelementary function, 106 Nonhomogeneous equation, 266, 287 Nonhomogeneous system, 248, 370, 382, 482 Nonlinear pendulum, 544 Nonlinear spring, 539 Nonlinear vibrations, 542 Nonsingular matrix, 183 properties, 183 Null space, 255 and row space, 255 O On-off function, 577 One-parameter family of solutions, Operation (elementary), 149 Operator, polynomial differential, 293 Orbit (of comet), 467 Order of differential equation, Ordinary differential equation, Ordinary point, 616 Orthogonal basis, 253 Orthogonal complement, 254 properties of, 254 Orthogonal matrix, 202 Orthogonal vectors, 252 Orthogonality and linear independence, 253 Oscillating populations, 528 Oscillating springs: 737 hard, 540 soft, 541 Overdamping, 307 P Parallelogram law of addition, 212, 223 Parameter, 141 Parameters, variation of, 322 for linear systems, 485 Partial-fraction decomposition, 260, 579 Pendulum, 303, 311 nonlinear, 544 Period, 305, 547, 597 Periodic function, 597 Permutation, 668 Phase angle, 305 Phase diagram, 88 Phase plane, 369, 503 position-velocity, 539 Phase plane portrait, 369, 505 Physical units, 13 Picard, Emile (1856–1941), 654 Piecewise continuous function, 561 jump, 562 Piecewise smooth function, 568 Pivot columns of a matrix, 245 Polar form of a complex number, 298 Polking, John, 28, 513, 525 Polluted reservoir, 55 Polynomial, 203, 237, 259 interpolating, 204 Polynomial differential operator, 293 Population equation, 76 Population explosion, 76, 359 Population growth, 3, 35 Population vector, 356 Position function, 11 Position vector, 373 Potential energy, 303, 331 Power series, 604 Power series method, 604, 608 Power series representation, 605 Practical resonance, 333 Predation, 534 Predator-prey model (or situation or system), 357, 527, 535 Predictor-corrector methods, 120 Principal diagonal, 162 Principia Mathematica, 95, 373 Principle of superposition: for linear differential equations, 267, 279 for systems, 377 Product: matrix, 168 738 Index matrix with number, 165 of determinants, 196 Projection matrix, 491, 498 Proper node, 401, 506 Proper subspace, 219 Pseudofrequency, 308 Pseudoperiod, 308 Pythagorean formula, 253 Python programming language, 115, 125, 135, 465 Q Quadratic factor partial fraction, 579 Quadratic factors, repeated, 583 R R2 as a vector space, 214 R3 as a vector space, 214 Rabbit-fox example, 357 Radioactive decay, 36 Radius of convergence, 611 Railway car examples, 428, 446 Rank and dimension, 248 Rank r generalized eigenvector, 442 Rayleigh, Lord (John William Strutt, 1842–1919), 551 Recurrence relation, 609 many-term, 620 two-term, 620 Reduced echelon matrix, 157 uniqueness of, 158 Reducible second-order equation: dependent variable missing, 68 independent variable missing, 69 Reduction of order, 289 Regular singular point, 629 Repeated eigenvalue, 399 negative, 408 positive, 405 zero, 409 Resistance matrix, 446 Resonance, 330, 432 and repeated quadratic factors, 583 practical, 333 Riccati equation, 70, 651 River crossing, 15 Rocket propulsion equation, 104 Rodrigues’ formula, 624 Rolling disk, 331 Roots of characteristic equation: complex, 297 distinct real, 274, 292 repeated complex, 299 repeated real, 275, 295, 405 Rota, Gian-Carlo (1932–1999), 118 Roundoff error, 111 Row (of a matrix), 147 Row equivalent matrices, 150 Row operations, 149 and determinants, 192 and elementary matrices, 180 and invertible matrices, 181 Row rank, 243 and column rank, 247 Row space, 243 and null space, 255 basis for, 244 of an echelon matrix, 243 of equivalent matrices, 243 Row vector, 166, 169 of a matrix, 169, 242 Runge, Carl (1856–1927), 127 Runge-Kutta method, 128 error in, 128 for systems, 456 variable step size methods, 460 S Saddle point, 400, 507 Sawtooth function, 577 Second law of motion, 11, 13, 94, 95, 266, 302, 327, 365, 425 Second-order equation, 68 approximate solution, 456 Second-order system, 426, 496 Separable differential equation, 32 Separation of variables, 31 Separatrix, 532, 541 Series: binomial, 605 convergence, 605 exponential, 606 formal multiplication, 606 geometric, 605 hypergeometric, 640 identity principle, 609 Maclaurin, 606 power, 604 radius of convergence, 611 shift of index, 610 Taylor, 606 termwise addition, 606 termwise differentiation, 608 Shape (of a matrix), 147 Shift of index, 610 Similar matrices, 349 Simple harmonic motion, 305 Simple pendulum, 303 Sine integral function, 50 Singular matrix, 183 Singular point, 616 irregular, 629 regular, 629 Singular solution, 35 Sink, 401, 408, 507 Size (of a matrix), 147 Skydiver, 131, 136 Skywalk, 330 Slope field, 18, 505 applications of, 20 Snowplow problem, 43 Soft spring, 541 Soft touchdown, 12 Solar wind, 17 Solution: eigenvalue, 386 equilibrium, 49, 88, 504 existence, uniqueness, 23, 371, 662, 663 fundamental matrix, 470 general, 10, 11, 35, 271, 272, 286 implicit, 34 linear first-order, 47 of differential equation, of linear (algebraic) system, 139 of nonhomogeneous equation, 287 of nonhomogeneous system, 382, 482 of second-order system, 427, 496 of system of differential equations, 365, 371, 376 one-parameter family, particular, 10 singular, 35 straight-line, 403 trivial, 160 two-parameter family, zero, 403 Solution curve, 18, 369 Solution near ordinary point, 617 Solution set, 139 Solution space: basis, 240 of a linear (algebraic) system, 226 of a linear differential equation, 261, 267 Solution subspace, 226 Solutions in Bessel functions, 649 Solve, an initial value problem, Source, 402, 406, 507 Spacecraft landing, 464 Span, spanning set, 229, 238 unique, 232, 242 Spectral decomposition methods, 490 Spectral decomposition of a matrix, 493, 499 Spiral point (sink or source),415, 509 Spout, 89 Spring constant, 302 Square matrix, 161 Index Square wave function, 566, 577 Stability: of almost linear systems, 519 of linear systems, 518 Stable center, 508 Stable critical point, 89, 507 Staircase function, 566, 576 Standard basis for Rn , 235 Standard unit vectors, 231 Star, see Proper node Static displacement, 303, 330 Static equilibrium position, 303 Steady periodic solution, 433 Step function, 562, 594 Step size, 106 Stiffness matrix, 425 Stirling’s approximation, 55 Stochastic matrix, 357 Stokes’ drag law, 313 Stonehenge, 38 Straight-line solution, 403 Subspace, 224 criterion, 225 intersection, 228 of R3 , 219 proper, 226 solution, 226 sum, 228 zero, 226 Substitution method, 58 Sum: of matrices, 164 of subspaces, 228 of vectors, 212, 222 Superposition principle, see Principle of superposition Swimmer’s problem, 14 Systems of dimension two, 398 Trace (of a matrix), 347 Trajectory, 369, 503 closed, 509 Transform of derivative, 568 Transform of integral, 574 Transform of periodic function, 597 Transform perspective, 572 Transient solution, 332, 433 Transition matrix, 356 Translated series solutions, 619 Transpose (of a matrix), 195 Transposition, 668 Triangle inequality, 253 Triangle law of addition, 212 Triangular matrix, 194 Triangular wave function, 577 Trivial solution, 160 Two independent eigenvectors, 405 U Undamped forced oscillations, 327 Undamped motion, 303 Underdamping, 308 Undetermined coefficients, 314, 483 case of duplication, 319 Unicycle model of car, 331 Uniform convergence, 659 Unique solution, 140 Uniqueness of linear combinations, 232 Uniqueness of solutions, see Existence Uniquely spanning set, 232, 242 Unit step function, 562, 594 Unit vector, 176, 218 standard, 231 Units (physical), 13 Unstable critical point, 89, 507 Upper triangular matrix, 194 U S population, 78, 86 T Taylor series, 606 Temperature oscillations, 56 Terminal speed, 96 Termwise addition of series, 606 Termwise differentiation of series, 608 Three possibilities (for linear systems), 139, 160 Threshold population, 81, 90 Time lag, 305 Time reversal, 402 Time-varying amplitude, 308 Torricelli’s law, 2, 39 V van der Pol, Balthasar (1889–1959), 552 van der Pol’s equation, 552 Vandermonde determinant, 204, 289 Vandermonde matrix, 204, 260 Variable gravitational acceleration, 99 Variables: free, 151, 239 leading, 151, 239 Variation of parameters, 322, 485 formula, 324, 486 739 Vector: 2-vector, 214 3-vector, 212 basic unit, 176, 218 column, 148, 169 component, 212 coordinates, 211 length, 213, 251 multiplication by scalar, 213, 223 n-vector, 222 norm, 658 orthogonal, 252 row, 166, 169 scalar product, 167 sum, 212, 222 unit, 176, 218 zero, 213, 223 Vector addition, 212, 222 Vector space, 223 basis of, 235 dimension, 237 finite-dimensional, 237 infinite dimensional, 237 of functions, 224, 259 subspace, 224 Velocity, 11 limiting, 20 Verhulst, Pierre-Franc¸ois (1804–1849), 78 Verne, Jules (1828–1905), 100, 103 Vertical motion with gravitational acceleration, 13 with air resistance, 95 Viscosity, 313 Volterra, Vito (1860–1940), 526 W Watson, G N (1886–1965), 642 Weight, 13 Well-posed problems and mathematical models, 665 Without two independent eigenvectors, 406 World population, 37, 87, 205 Wronskian, 271, 283 of solutions, 272, 285, 289 of vector-valued functions, 378 Z Zero matrix, 172 Zero subspace, 226 Zero vector, 213, 223 Table of Laplace Transforms This table summarizes the general properties of Laplace transforms and the Laplace transforms of particular functions derived in Chapter 10 Function Transform Function f t / F s/ e at af t / C bg.t / aF s/ C bG.s/ t n e at f t / sF s/ cos kt f 00 t / s F s/ sf 0/ f n/ t / s n F s/ sn Z t e at f t / Z a/f t f 0/ f 0/


F s/ s f / d  u.t f 0/ F s as F s/ e / d  F s/G.s/ 1/n F n/ s/ f t / t Z period p u.t a/ e ps ı.t a/ s a a/2 C k s k a/2 C k s C k /2 s s C k /2 s e e st f t / dt t s2   t a p t p s ta €.a C 1/ s aC1 as s e as 1/Œ t=a nŠ s2 C k /2 p s s nC1 k2 s s tn kt cos kt/ Z k2 k sin kt C kt cos kt/ 2k F  / d s C k2 s2 t sin kt 2k t n f t / nŠ a/nC1 s2 sin kt 2k F s/ s s cosh kt e at sin kt f t /, 1/ 0/ a k s2 C k2 sin kt f n s s2 e at cos kt a/ a/ tf t / sinh kt t f /g.t Transform (square wave) (staircase) as s e s.1 as e as / Table of Integrals ELEMENTARY FORMS Z Z u dv D uv v du 10 Z sec u tan u du D sec u C C 11 Z csc u cot u du D 12 Z tan u du D ln j sec uj C C 13 Z cot u du D ln j sin uj C C sec u du D ln j sec u C tan uj C C Z u du D unC1 C C nC1 Z du D ln juj C C u Z e du D e C C Z au a du D CC ln a 14 Z Z sin u du D 15 Z csc u du D ln j csc u Z cos u du D sin u C C 16 Z du p D sin a u2 Z sec2 u du D tan u C C 17 Z a2 Z csc2 u du D 23 Z sin3 u du D 24 Z cos3 u du D C cos2 u/ sin u C C 25 Z tan3 u du D 26 Z cot3 u du D n u if n 6D u u cos u C C TRIGONOMETRIC FORMS Z 1 19 sin2 u du D u sin 2u C C Z 1 20 cos2 u du D u C sin 2u C C Z 21 tan2 u du D tan u u C C 22 Z cot2 u du D cot u uCC 1 sec u tan u C ln j sec u C tan uj C C 2 27 Z sec3 u du D 28 Z csc3 u du D 29 Z sin au sin bu du D 1 csc u cot u C ln j csc u 2 sin.a 2.a cot uj C C u CC a u CC a ˇ ˇ Z ˇu C aˇ du ˇ ˇCC 18 D ln a u2 2a ˇ u a ˇ cot u C C b/u b/ du D tan Cu a csc u C C 1 C sin2 u/ cos u C C tan2 u C ln j cos uj C C cot2 u ln j sin uj C C cot uj C C sin.a C b/u CC 2.a C b/ if a2 6D b (Continued on Rear Endpaper) Table of Integrals (cont.) sin.a C b/u b/u C CC b/ 2.a C b/ sin.a 2.a 30 Z 31 Z 32 Z 33 Z 34 Z 35 Z 36 Z 37 Z 38 Z u sin u du D sin u 39 Z u cos u du D cos u C u sin u C C 40 Z u sin u du D 41 Z un cos u du D un sin u cos au cos bu du D if a2 6D b cos.a C b/u C C if a2 6D b 2.a C b/ Z n sinn u du D sinn u cos u C sinn u du n n Z n n n cos u du D cos u sin u C cosn u du n n Z tann u du D tann u tann u du if n 6D n Z n n cot u cotn u du if n 6D cot u du D n Z n secn u tan u C secn u du if n 6D secn u du D n n Z n cscn u du if n 6D cscn u du D cscn u cot u C n n sin au cos bu du D n cos.a b/u 2.a b/ u cos u C C n u cos u C n n Z Z un un 1 cos u du sin u du p FORMS INVOLVING u2 ˙ a2 Z p ˇ p up a2 ˇˇ ˇ 42 u2 ˙ a2 du D u ˙ a2 ˙ ln ˇu C u2 ˙ a2 ˇ C C 2 Z ˇ ˇ p du ˇ ˇ 43 p D ln ˇu C u2 ˙ a2 ˇ C C u2 ˙ a p FORMS INVOLVING a2 u2 Z p up a2 u 44 a2 u2 du D a u2 C sin C C 2 a ˇ ˇ Z p ˇ a C pa2 u2 ˇ p a u2 ˇ ˇ 2 45 du D a u a ln ˇ ˇCC ˇ ˇ u u Table of Integrals (cont.) EXPONENTIAL AND LOGARITHMIC FORMS Z 46 ue u du D u 1/e u C C 47 Z un e u du D un e u 48 Z unC1 un ln u du D ln u nC1 n Z un u e du Z u sin u du D 2u2 55 Z u tan u du D u C 1/ tan 56 Z u sec u du D u2 sec 57 Z un sin u du D unC1 sin nC1 58 Z un tan u du D unC1 tan nC1 59 Z u sec 62 =2 sinn u du D Z =2 uC Z e au cos bu du D Z tan e au a cos bu C b sin bu/ C a2 C b up 52 u du D u tan u ln.1 C u2 / C C u2 C C 1CC u nC1 Z unC1 p du u2 u nC1 Z unC1 du C u2 u nC1 Z unC1 p du u2 1 u CC u 1p u u OTHER USEFUL FORMULAS Z 60 un e u du D €.n C 1/ D nŠ Z 1/ sin unC1 u du D sec nC1 50 C 54 e au sin bu du D unC1 CC n C 1/2 INVERSE TRIGONOMETRIC FORMS Z p 51 sin u du D u sin u C u2 C C Z ˇ ˇ p ˇ ˇ 53 sec u du D u sec u ln ˇu C u2 1ˇ C C n e au a sin bu b cos bu/ C C a2 C b 49 Z (n = 0) 61 if n 6D if n 6D Z if n 6D 1 e




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n au2 du D r  a (a > 0) if n is an even integer and n = if n is an odd integer and n = ... elementary linear algebra concepts and techniques that are needed for the solution of linear differential equations and systems Chapter includes sections 4.5 (row and column spaces) and 4.6 (orthogonal... differential equations and linear algebra in tandem reinforces the learning of both subjects In this book we therefore have combined core topics in elementary differential equations with those concepts and. .. elementary linear algebra that are needed for a contemporary introduction to differential equations Principal Features of This Revision This 4th edition is the most comprehensive and wide-ranging