Advanced Engineering Mathematics EIGHTH EDITION Advanced Engineering Mathematics ERWIN KREYSIIG Professor of Mathematics Ohio State University Columbus, Ohio JOHN WILEY & SONS, INe Publisher: Peter Janzow Mathematics Editor: Barbara Hol/and Marketing Manager: Audra Silveric Freelance Production Manager: Jeanine Furino Lorraine Burke HRS Electronic Text Management Designer: HRS Electronic Text Management Illustration Editor: Sigmund Malinowski Electronic Illustrations: Precision Graphies Cover Photo: Chris Rogers/The Stock Market This book was set in Times Roman by GGS Information Services and printed and bound by Von Hoffmann Press The coyer was printed by Phoenix Color Corp This book is printed on acid-free paper Copyright © 1999 John Wiley & Sons, Ine Ail rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978)750-4470 Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Ine., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM To order books please cali 1(800)-225-5945 Kreyszig, Erwin Advanced engineering mathematics / Erwin Kreyszig.-8th ed p cm Accompanied by instructor's manual Includes bibliographical references and index ISBN 0-471-15496-2 (cloth : acid-free paper) Mathematical physics Engineering mathematics Title Printed in the United States of America 10 See also http://www.wiley.com/college/mat/kreyszig154962/ Purpose of the Book This book introduces students of engineering, physics, mathematics, and computer science to those areas of mathematics which, from a modem point of view, are most important in connection with practical problems The content and character of mathematics needed in applications are changing rapidly Linear algebra-especially matrices-and numerical methods for computers are of increasing importance Statistics and graph theory play more prominent roles Real analysis (ordinary and partial differential equations) and complex analysis remain indispensable The material in this book is arranged accordingly, in seven independent parts (see also the diagram on the next page): A B C D E F G Ordinary DifferentiaI Equations (Chaps 1- 5) Linear Algebra, Vector Calculus (Chaps 6-9) Fourier Analysis and Partial DifferentiaI Equations Complex Analysis (Chaps 12-16) Numerical Methods (Chaps 17 -19) Optimization, Graphs (Chaps 20, 21) Probability and Statistics (Chaps 22, 23) (Chaps 10, Il) This is followed by References (Appendix 1) Answers to Odd-Numbered Problems (Appendix 2) Auxiliary Material (Appendix and inside of covers) Additional Proofs (Appendix 4) Tables of Functions (Appendix 5) This book has helped to pave the way for the present development and will prepare students for the present situation and the future by a modem approach to the areas listed above and the ide as-sorne of them computer related-that are presently causing basic changes: Many methods have become obsolete New ideas are emphasized, for instance, stability, error estimation, and structural problems of algorithms, to mention just a few Trends are driven by supply and demand: supply of powerful new mathematical and computational methods and of enormous computer capacities, demand to solve problems of growing complexity and size, arising from more and more sophisticated systems or production processes, from extreme physical conditions (for example, those in space travel), from materials with unusual properties (plastics, alloys, superconductors, etc.), or from entirely new tasks in computer vision, robotics, and other new fields The general trend seems c1ear Details are more difficult to predict Accordingly, students need solid knowledge of basic principles, methods, and results, and a c1ear perception of what engineering mathematics is aIl about, in aIl three phases of solving problems: • Modeling: Translating given physical or other information and data into mathematical form, into a mathematical model (a differential equation, a system of equations, or sorne other expression) Preface vii • Solving: Obtaining the solution by selecting and applying suitable mathematical methods, and in most cases doing numerical work on a computer This is the main task of this book • Interpreting: Understanding the meaning and the implications of the mathematical solution for the original problem in terms of physics ûr wherever the problem cornes from It would make no sense to overload students with aIl kinds of little things that might be of occasion al use Instead, it is important that students become familiar with ways to think mathematically, recognize the need for applying mathematical methods to engineering problems, realize that mathematics is a systematic science built on relatively few basic concepts and involving powerful unifying principles, and get a firrn grasp for the interrelation between theory, computing, and experiment The rapid ongoing developments just sketched have led to many changes and new features in the present edition of this book ln particular, many sections have been rewritten in a more detailed and leisurely fashion to make it a simpler book This has also led to a still better balance between applications, algorithmic ideas, worked-out examples, and theory Big Changes in This Edition PROBLEM SETS CHANGED The new problems place more emphasis on qualitative methods and applications There is a (slight) reduction of formaI manipulations in favor of problems that require mathematical thinking and understanding, as opposed to a routine use of a CAS (Computer Algebra System) PROJECTS Modern engineering work is team work, and TEAM PROJECTS will help the student to prepare for this (These are relatively simple, so that they will fit into the time schedule of a busy student.) WRITING PROJECTS will help in learning how to plan, develop, and write coherent reports CAS PROJECTS and CAS PROBLEMS will invite the student to an increased use of computers (and programmable cakulators); these projects are not mandatory, simply because this book can be used independently of computers or in connection with them (see page x) NUMERICAl ANAlYSIS UPDATED Details are given below Further Changes and New Features in Chapters Ordinary Differentiai Equations (Chaps 1-5) ~ First-Order Differentiai Equations (Chap 1) Qualitative aspects emphasized by discussing direction fields early (Sec 1.2) Presentation streamlined by combining exact equations and integrating factors into one section (Sec 1.5) and moving Picard's iteration to Sec 1.9 on existence and uniqueness ix Preface Suggestions for Courses: A Four-Semester Sequence The material may be taken in sequence and is suitable for four consecutive courses, meeting - hours a week: First semester Second semester Third semester Fourth semester semester Ordinary differential equations (Chaps 1-4 or 5) Linear algebra and vector analysis (Chaps 6-9) Complex analysis (Chaps 12-16) Numerical methods (Chaps 17-19) For the remaining chapters, see below Possible interchanges are obvious; for instance, numerical methods could precede complex analysis, etc Suggestions for Courses: Independent One-Semester Courses The book is also suitable for various independent one-semester courses meeting hours a week; for example: Introduction to ordinary differential equations (Chaps 1- 2) Laplace transform (Chap 5) Vector algebra and caIculus (Chaps 8, 9) Matrices and linear systems of equations (Chaps 6, 7) Fourier series and partial differential equations (Chaps 10, Il, Secs 19.4-19.7) Introduction to complex analysis (Chaps 12-15) Numerical analysis (Chaps 17, 19) Numericallinear algebra (Chap 18) Optimization (Chaps 20, 21) Graphs and combinatorial optimization (Chap 21) Probability and statistics (Chaps 22, 23) General Features of This Edition The selection, arrangement, and presentation of the material has been made with greatest care, based on past and present teaching, research, and consulting experience Sorne major features of the book are these: The book is self-contained, except for a few c1early marked places where a proof would be beyond the level of a book of the present type and a reference is given instead Hiding difficulties or oversimplifying would be of no real help to students The presentation is detailed, to avoid irritating readers by frequent references to details in other books The examples are simple, to make the book teachable-why choose complicated examples when simple ones are as instructive or ev en better? x Preface The notations are modern and standard, ta help students read articles in joumals or other modern books and understand other mathematically oriented courses The chapters are largely independent, providing flexibility in teaching special courses (see above) Computer Use and CASs (Computer Aigebraic Systems) The use of computers and (programmable) calculators is invited but not requested This technology may be helpful in solving many of the about 4000 problems in this book Intelligent utilization of these amazingly powerful and versatile systems may give the student additional motivation, insight, and help in working in classes, tutorials, labs, and at home, as weIl as in the preparation for future jobs after graduation For these reasons we have included CAS projects as useful enrichments of the problem sets, which, however, remain complete entities without them Similarly, working through the text is possible without computer use A list of software is included before Chap 17, the first chapter on numerical methods, on p 829 Acknowledgments am indebted to many of my former teachers, colleagues, and students who directly or indirectly helped me in preparing this book, in particular, the present edition of it Various parts of the manuscript were copied and distributed to my classes and retumed to me with suggestions for improvement Discussions with engineers and mathematicians (as weIl as written comments) were of great help to me; want to mention particularly Professors S L Campbell, J T Cargo, R Carr, P L Chambré, V F ConnoIly, J Delany, J W Dettman, D Dicker, L D Drager, D Ellis, W Fox, R B Guenther, J L Handley, V W Howe, W N Huff, J Keener, V Komkow, H Kuhn, G Lamb, H B Mann, Marx, K Millet, D Moore, W D Munroe, A Nadim, N Ong, Jr., P J Pritchard, W O Ray, J T Scheick, L F Shampine, H A Smith, J Todd, H Unz, A L Villone, H J Weiss, A Wilansky, C H Wilcox, H Ya Fan, L Zia, A D Ziebur, aIl from the U.S.A., Professors H S M Coxeter, E.1 Norminton, R Vaillancourt, and Mr H Kreyszig (whose computer expertise was of great help in Chaps 17 -19) from Canada, and Professors H Florian, H Unger, and H Wielandt from Europe can offer here only an inadequate acknowledgment of my appreciation My very special cordial thanks goes to Privatdozent Dr M Kracht for the formidable task of checking aIl the details of the manuscript, resulting in many substantial improvements also want to thank Ms Barbara Holland, Editor, for her unusually great help and effort during the periods of preparing the manuscript and producing the book Furthermore, wish to thank John Wiley and Sons (see the list on p iv) as weIl as Mr J T Nystrom and Ms C A Elicker of GGS Information Services for their effective cooperation and great care in preparing this edition Suggestions of many readers were evaluated in preparing the present edition Any further comments and suggestions for improvement of the book will be gratefully received ERWIN KREYSZIG Part A Ordinary Differentiai Equations Chapter First-Order Differentiai Equations Basic Concepts and Ideas, 1.1 Geometrical Meaning of y = f(x, y) Direction Fields, 10 1.2 1.3 Separable DifferentiaI Equations, 14 1.4 Modeling: Separable Equations, 19 Exact DifferentiaI Equations Integrating Factors, 25 1.5 Linear DifferentiaI Equations Bernoulli Equation, 33 1.6 Modeling: Electric Circuits, 41 1.7 1.8 Orthogonal Trajectories of Curves Optional, 48 Existence and Uniqueness of Solutions Picard Iteration, 52 1.9 Chapter Review, 59 Chapter Summary, 61 Chapter Linear Differentiai Equations of Second and Higher Order 64 Homogeneous Linear Equations of Second Order, 64 2.1 Second-Order Homogeneous Equations with Constant Coefficients, 72 2.2 Case of Complex Roots Complex Exponential Function, 76 2.3 2.4 DifferentiaI Operators Optional, 81 Modeling: Free Oscillations (Mass-Spring System), 83 2.5 Euler-Cauchy Equation, 93 2.6 Existence and Uniqueness Theory Wronskian, 97 2.7 Nonhomogeneous Equations, 101 2.8 Solution by Undetermined Coefficients, 104 2.9 2.10 Solution by Variation of Parameters, 108 2.11 Modeling: Forced Oscillations Resonance, III 2.12 Modeling of Electric Circuits, 118 2.13 Higher Order Linear DifferentiaI Equations, 124 2.14 Higher Order Homogeneous Equations with Constant Coefficients, 132 2.15 Higher Order Nonhomogeneous Equations, 138 Chapter Review, 142 Chapter Summary, 143 Chapter Systems of Differentiai Equations, Phase Plane, Qualitative Methods 146 Introduction: Vectors, Matrices, Eigenvalues, 146 Introductory Examples, 152 Basic Concepts and Theory, 159 Homogeneous Systems with Constant Coefficients Phase Plane, Critical Points, 162 Criteria for Critical Points Stability, 170 3.4 Qualitative Methods for Nonlinear Systems, 175 3.5 Nonhomogeneous Linear Systems, 184 3.6 Chapter Review, 190 Chapter Summary, 192 3.0 3.1 3.2 3.3 xi 3.4 Criteria for Critical Points Stability In the preceding section we discussed homogeneous linear systems in two equations with constant coefficients See 3.4 Criteria for Critical Points Stability 171 constant coefficients ajk in (1) this leads to eigenvalue problems The simple reason is that we set 174 Systems of Differential Solution The equation is [see (5) in Sec 2.5] Equations Chap We see that these three terms are energies Indeed, Y2 is the angular velocity, so that LY2 is the velocity and the first term is the kinetic energy The second term (including the minus sign) is the potential energy of the pendulum, and mL2C is its total energy, which is constant, as expected from the law of conservation of energy, because there is no damping (no loss of energy) The type of motion depends on the total energy, hence on C, as follows Systems of Differential Equations 184 3.6 Nonhomogeneous Chap Linear Systems In this last section of Chap we discuss methods for solving nonhomogeneous systems From Sec 3.2 we recall that such a system is of the form linear ...EIGHTH EDITION Advanced Engineering Mathematics ERWIN KREYSIIG Professor of Mathematics Ohio State University Columbus, Ohio JOHN WILEY & SONS, INe Publisher: Peter Janzow Mathematics Editor:... E-Mail: PERMREQ@WILEY.COM To order books please cali 1(800)-225-5945 Kreyszig, Erwin Advanced engineering mathematics / Erwin Kreyszig.-8th ed p cm Accompanied by instructor's manual Includes... physics Engineering mathematics Title Printed in the United States of America 10 See also http://www.wiley.com/college/mat/kreyszig154962/ Purpose of the Book This book introduces students of engineering,