Advanced Engineering Mathematics

753 Pages·2019·7.242 MB·English

by  Merle C. Potter, Jack. L. Lessing, Edward F. Aboufadel

Merle C. Potter
Jack L. Lessing
Edward F. Aboufadel
Advanced Engineering
Mathematics
Fourth Edition
Advanced Engineering Mathematics
Merle C. Potter (cid:129) Jack L. Lessing
Edward F. Aboufadel
Advanced Engineering
Mathematics
Fourth Edition
Merle C. Potter Jack L. Lessing
Mechanical Engineering University of Michigan
Michigan State University Ft. Lauderdale, FL, USA
Ann Arbor, MI, USA

Edward F. Aboufadel
Grand Valley State University
Ada, MI, USA
ISBN 978-3-030-17067-7 ISBN 978-3-030-170 68-4 (eB ook)
https://doi.org/10.1007/978-3-030-17068-4

© Springer Nature Switzerland AG 2019
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or
part of the material is concerned, specifically the rights of translation, reprinting, reuse of
illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way,
and transmission or information storage and retrieval, electronic adaptation, computer software,
or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in
this book are believed to be true and accurate at the date of publication. Neither the publisher
nor the authors or the editors give a warranty, expressed or implied, with respect to the material
contained herein or for any errors or omissions that may have been made. The publisher remains
neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG.
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
The purpose of this book is to introduce students of the physical sciences to several mathemati-
cal methods often essential to the successful solution of real problems. The methods chosen are
those most frequently used in typical physics and engineering applications. The treatment is not
intended to be exhaustive; the subject of each chapter can be found as the title of a book that
treats the material in much greater depth. The reader is encouraged to consult such a book should
more study be desired in any of the areas introduced.
Perhaps it would be helpful to discuss the motivation that led to the writing of this text.
Undergraduate education in the physical sciences has become more advanced and sophisticated
with the advent of the space age and computers, with their demand for the solution of very diffi-
cult problems. During the recent past, mathematical topics usually reserved for graduate study
have become part of the undergraduate program. It is now common to find an applied mathe-
matics course, usually covering one topic, that follows differential equations in engineering and
physical science curricula. Choosing the content of this mathematics course is often difficult. In
each of the physical science disciplines, different phenomena are investigated that result in a
variety of mathematical models. To be sure, a number of outstanding textbooks exist that present
advanced and comprehensive treatment of these methods. However, these texts are usually writ-
ten at a level too advanced for the undergraduate student, and the material is so exhaustive that
it inhibits the effective presentation of the mathematical techniques as a tool for the analysis of
some of the simpler problems encountered by the undergraduate. This book was written to pro-
vide for an additional course, or two, after a course in differential equations, to permit more than
one topic to be introduced in a term or semester, and to make the material comprehensive to the
undergraduate. However, rather than assume a knowledge of differential equations, we have
included all of the essential material usually found in a course on that subject, so that this text
can also be used in an introductory course on differential equations or in a second applied course
on differential equations. Selected sections from several of the chapters would constitute such
courses.
Ordinary differential equations, including a number of physical applications, are reviewed in
Chapter 1. The use of series methods is presented in Chapter 2. Subsequent chapters present
Laplace transforms, matrix theory and applications, vector analysis, Fourier series and trans-
forms, partial differential equations, numerical methods using finite differences, complex vari-
ables, and wavelets. The material is presented so that more than one subject, perhaps four sub-
jects, can be covered in a single course, depending on the topics chosen and the completeness of
coverage. The style of presentation is such that the reader, with a minimum of assistance, may
follow the step-by-step derivations from the instructor. Liberal use of examples and homework
problems should aid the student in the study of the mathematical methods presented.
Incorporated in this new edition is the use of certain computer software packages. Short tuto-
rials on Maple, demonstrating how problems in advanced engineering mathematics can be
solved with a computer algebra system, are included in most sections of the text. Problems have
been identified at the end of sections to be solved specifically with Maple, and there are also
computer laboratory activities,which are longer problems designed for Maple. Completion of
these problems will contribute to a deeper understanding of the material. There is also an ap-
pendix devoted to simple Maplecommands. In addition, Matlaband Excelhave been included
v
vi (cid:1) PREFACE
in the solution of problems in several of the chapters. Excelis more appropriate than a computer
algebra system when dealing with discrete data (such as in the numerical solution of partial dif-
ferential equations).
At the same time, problems from the previous edition remain, placed in the text specifically
to be done without Maple. These problems provide an opportunity for students to develop and
sharpen their problem-solving skills—to be human algebra systems.1 Ignoring these sorts of
exercises will hinder the real understanding of the material.
The discussion of Maplein this book uses Maple 8, which was released in 2002. Nearly all
the examples are straightforward enough to also work in Maple 6, 7, and the just-released 9.
Maplecommands are indicated with a special input font, while the output also uses a special font
along with special mathematical symbols. When describing Excel,the codes and formulas used
in cells are indicated in bold, while the actual values in the cells are not in bold.
Answers to numerous even-numbered problems are included just before the Index, and a
solutions manual is available to professors who adopt the text. We encourage both students and
professors to contact us with comments and suggestions for ways to improve this book.
Merle C. Potter
Jack. L. Lessing
Edward F. Aboufadel
1We thank Susan Colley of Oberlin College for the use of this term to describe people who derive formulas and
calculate answers using pen and paper.
CONTENTS
Preface v
1 Ordinary Differential Equations 1
1.1 Introduction 1
1.2 Definitions 2
1.2.1 MapleApplications 5
1.3 Differential Equations of First Order 7
1.3.1 Separable Equations 7
1.3.2 MapleApplications 11
1.3.3 Exact Equations 12
1.3.4 Integrating Factors 16
1.3.5 MapleApplications 20
1.4 Physical Applications 20
1.4.1 Simple Electrical Circuits 21
1.4.2 MapleApplications 23
1.4.3 The Rate Equation 23
1.4.4 MapleApplications 25
1.4.5 Fluid Flow 26
1.5 Linear Differential Equations 28
1.5.1 Introduction and a Fundamental Theorem 28
1.5.2 Linear Differential Operators 31
1.5.3 Wronskians and General Solutions 33
1.5.4 Maple Applications 35
1.5.5 The General Solution of the Nonhomogeneous
Equation 36
1.6 Homogeneous, Second-Order, Linear Equations
with Constant Coefficients 38
1.6.1 MapleApplications 42
1.7 Spring–Mass System: Free Motion 44
1.7.1 Undamped Motion 45
1.7.2 Damped Motion 47
1.7.3 The Electrical Circuit Analog 52
1.8 Nonhomogeneous, Second-Order, Linear Equations
with Constant Coefficients 54
vii
viii (cid:1) CONTENTS
1.9 Spring–Mass System: Forced Motion 59
1.9.1 Resonance 61
1.9.2 Near Resonance 62
1.9.3 Forced Oscillations with Damping 64
1.10 Variation of Parameters 69
1.11 The Cauchy–Euler Equation 72
1.12 Miscellania 75
1.12.1 Change of Dependent Variables 75
1.12.2 The Normal Form 76
1.12.3 Change of Independent Variable 79
Table 1.1 Differential Equations 82
2 Series Method 85
2.1 Introduction 85
2.2 Properties of Power Series 85
2.2.1 MapleApplications 92
2.3 Solutions of Ordinary Differential Equations 94
2.3.1 MapleApplications 98
2.3.2 Legendre’s Equation 99
2.3.3 Legendre Polynomials and Functions 101
2.3.4 MapleApplications 103
2.3.5 Hermite Polynomials 104
2.3.6 MapleApplications 105
2.4 The Method of Frobenius: Solutions About
Regular Singular Points 106
2.5 The Gamma Function 111
2.5.1 MapleApplications 115
2.6 The Bessel–Clifford Equation 116
2.7 Laguerre Polynomials 117
2.8 Roots Differing by an Integer: The Wronskian Method 118
2.9 Roots Differing by an Integer: Series Method 122
2.9.1 s=0 124
2.9.2 s=N, N a Positive Integer 127
2.10 Bessel’s Equation 130
2.10.1 Roots Not Differing by an Integer 131
2.10.2 MapleApplications 133
2.10.3 Equal Roots 134
2.10.4 Roots Differing by an Integer 136
2.10.5 MapleApplications 137
2.10.6 Basic Identities 138
2.11 Nonhomogeneous Equations 142
2.11.1 MapleApplications 146
CONTENTS (cid:1) ix
3 Laplace Transforms 147
3.1 Introduction 147
3.2 The Laplace Transform 147
3.2.1 MapleApplications 158
3.3 Laplace Transforms of Derivatives and Integrals 162
3.4 Derivatives and Integrals of Laplace Transforms 167
3.5 Laplace Transforms of Periodic Functions 171
3.6 Inverse Laplace Transforms: Partial Fractions 175
3.6.1 Unrepeated Linear Factor (s−a) 175
3.6.2 MapleApplications 176
3.6.3 Repeated Linear Factor (s−a)m 177
3.6.4 Unrepeated Quadratic Factor [(s−a)2+b2] 178
3.6.5 Repeated Quadratic Factor [(s−a)2+b2]m 180
3.7 A Convolution Theorem 181
3.7.1 The Error Function 183
3.8 Solution of Differential Equations 184
3.8.1 MapleApplications 192
3.9 Special Techniques 195
3.9.1 Power Series 195
Table 3.1 Laplace Transforms 198
4 The Theory of Matrices 200
4.1 Introduction 200
4.1.1 MapleApplications 200
4.2 Notation and Terminology 200
4.2.1 Maple, Excel,and MATLAB Applications 202
4.3 The Solution of Simultaneous Equations by
Gaussian Elimination 207
4.3.1 Mapleand MATLAB Applications 212
4.4 Rank and the Row Reduced Echelon Form 216
4.5 The Arithmetic of Matrices 219
4.5.1 Maple, Excel, and MATLAB Applications 222
4.6 Matrix Multiplication: Definition 225
4.6.1 Maple, Excel, and MATLAB Applications 229
4.7 The Inverse of a Matrix 233
4.8 The Computation of A−1 236
4.8.1 Maple, Excel, and MATLAB Applications 240
4.9 Determinants of n×nMatrices 243
4.9.1 Minors and Cofactors 249
4.9.2 Mapleand ExcelApplications 251
4.9.3 The Adjoint 252
x (cid:1) CONTENTS
4.10 Linear Independence 254
4.10.1 MapleApplications 258
4.11 Homogeneous Systems 259
4.12 Nonhomogeneous Equations 266
5 Matrix Applications 271
5.1 Introduction 271
5.1.1 Mapleand ExcelApplications 271
5.2 Norms and Inner Products 271
5.2.1 Mapleand MATLAB Applications 276
5.3 Orthogonal Sets and Matrices 278
5.3.1 The Gram–Schmidt Process and
the Q–RFactorization Theorem 282
5.3.2 Projection Matrices 288
5.3.3 Mapleand MATLAB Applications 291
5.4 Least Squares Fit of Data 294
5.4.1 Minimizing ||Ax−b|| 299
5.4.2 Mapleand ExcelApplications 300
5.5 Eigenvalues and Eigenvectors 303
5.5.1 Some Theoretical Considerations 309
5.5.2 Mapleand MATLAB Applications 312
5.6 Symmetric and Simple Matrices 317
5.6.1 Complex Vector Algebra 320
5.6.2 Some Theoretical Considerations 321
5.6.3 Simple Matrices 322
5.7 Systems of Linear Differential Equations:
The Homogeneous Case 326
5.7.1 Mapleand MATLAB Applications 332
5.7.2 Solutions with Complex Eigenvalues 337
5.8 Systems of Linear Equations: The Nonhomogeneous Case 340
5.8.1 Special Methods 345
5.8.2 Initial-Value Problems 349
5.8.3 MapleApplications 350
6 Vector Analysis 353
6.1 Introduction 353
6.2 Vector Algebra 353
6.2.1 Definitions 353
6.2.2 Addition and Subtraction 354
6.2.3 Components of a Vector 356
6.2.4 Multiplication 358
6.2.5 MapleApplications 366