ADVANCED ENGINEERING MATHEMATICS

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1 ADVANCED ENGINEERING MATHEMATICS DENNIS G. ZILL Loyola Marymount University MICHAEL R. CULLEN Loyola Marymount University PWS-KENT O I^7 3 PUBLISHING COMPANY E 9 U Boston

CONTENTS Preface xiii Parti ORDINARY DIFFERENTIAL EQUATIONS 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 3 1.1 1.2 1.

2 CONTENTS Preface xiii Parti ORDINARY DIFFERENTIAL EQUATIONS 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS Basic Definitions and Terminology 4 Differential Equation of a Family of Curves Mathematical Models 17 Summary 32 Chapter 1 Review Exercises 32 2 FIRST-ORDER DIFFERENTIAL EQUATIONS [O] 2.6 [O] 2.7 [O] Preliminary Theory 35 Separable Variables 39 Homogeneous Equations 46 Exact Equations 52 Linear Equations 58 Equations of Bernoulli, Ricatti, and Clairaut Substitutions 70 Picard's Method 73 Orthogonal Trajectories 76 Applications of Linear Equations 81 Applications of Nonlinear Equations 92 Summary 101 Chapter 2 Review Exercises 102 v

vi Contents 3 LINEAR DIFFERENTIAL EQUATIONS OF HIGHER ORDER 105 3.1 Preliminary Theory 106 3.1.1 Initial-Value and Boundary-Value Problems 106 3.1.2 Linear Dependence and Linear Independence 110 3.1.3 Solutions of Linear Equations 114 3.

3 vi Contents 3 LINEAR DIFFERENTIAL EQUATIONS OF HIGHER ORDER Preliminary Theory Initial-Value and Boundary-Value Problems Linear Dependence and Linear Independence Solutions of Linear Equations Constructing a Second Solution from a Known Solution Homogeneous Linear Equations with Constant Coefficients Undetermined Coefficients Differential Operators and Undetermined Coefficients Revisited Differential Operators 148 [O] An Alternative Approach to Undetermined Coefficients Variation of Parameters Systems of Linear Differential Equations with Constant Coefficients Simple Harmonie Motion Damped Motion Forced Motion Electric Circuits and Other Analogous Systems 198 Summary 204 Chapter 3 Review Exercises LAPLACE TRANSFORM Laplace Transform Inverse Transform Operational Properties Translation Theorems and Derivatives of a Transform Transforms of Derivatives and Integrals Transform of a Periodic Function Applications Dirac Delta Function Systems of Differential Equations 260 Summary 267 Chapter 4 Review Exercises DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS Cauchy-Euler Equation Power Series Solutions Solutions About Singular Points Two Special Equations 307

Contents vii 5.4.1 Solution of Bessel's Equation 307 5.4.2 Solution of Legendre's Equation 313 Summary 318 Chapter 5 Review Exercises 319 Part II VECTORS, MATRICES, AND VECTOR CALCULUS 321 6 VECTORS 323 6.

4 Contents vii Solution of Bessel's Equation Solution of Legendre's Equation 313 Summary 318 Chapter 5 Review Exercises 319 Part II VECTORS, MATRICES, AND VECTOR CALCULUS VECTORS Vectors in the Plane Vectors in Space The Dot Product The Cross Product Lines and Planes in 3-Space Vector Spaces 365 Summary 373 Chapter 6 Review Exercises MATRICES Matrix Algebra Systems of Linear Algebraic Equations Determinants Properties of Determinants Inverse of a Matrix Finding the Inverse Using the Inverse to Solve Systems Cramer's Ruie The Eigenvalue Problem Orthogonal Matrices Diagonalization Cryptography An Error-Correcting Code Method of Least Squares 465 Summary 469 Chapter 7 Review Exercises VECTOR CALCULUS Vector Functions Motion on a Curve; Velocity and Acceleration Curvature; Components of Acceleration Functions of Several Variables; Chain Ruie The Directional Derivative 502

viii Contents [0] [0] [0] 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.

5 viii Contents [0] [0] [0] Tangent Planes and Normal Lines 510 Divergence and Curl 515 Line Integrals 521 Line Integrals Independent of Path 533 Review of Double Integrals 542 Double Integrals in Polar Coordinates 553 Green's Theorem 559 Surface Integrals 566 Stokes' Theorem 575 Review of Triple Integrals 582 Divergence Theorem 597 Change of Variables in Multiple Integrals 605 Summary 613 Chapter 8 Review Exercises 614 Part III SYSTEMS OF DIFFERENTIAL EQUATIONS SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS Systems in Normal Form Matrix Functions Preliminary Theory Homogeneous Linear Systems Distinct Real Eigenvalues Complex Eigenvalues Repeated Eigenvalues Solution by Diagonalization Nonhomogeneous Linear Systems Undetermined Coefficients Variation of Parameters Diagonalization 664 [O] 9.7 Matrix Exponential 667 Summary 670 Chapter 9 Review Exercises PLANE AUTONOMOUS SYSTEMS AND STABILITY Autonomous Systems, Critical Points, and Periodic Solutions Stability and Linear Systems Linearization and Local Stability Applications of Autonomous Systems 698 [O] 10.5 Periodic Solutions, Limit Cycles, and Global Stability 707 Summary 718 Chapter 10 Review Exercises 720

Contents ix Part IV FOURIER SERIES AND BOUNDARY-VALUE PROBLEMS 723 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 725 11.1 Orthogonal Functions 726 11.2 Fourier Series 732 11.

6 Contents ix Part IV FOURIER SERIES AND BOUNDARY-VALUE PROBLEMS ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonal Functions Fourier Series Fourier Cosine and Sine Series Sturm-Liouville Problem Bessel and Legendre Series Fourier-Bessel Series Fourier-Legendre Series 759 Summary 762 Chapter 11 Review Exercises [O] 12.8 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 765 Separable Partial Differential Equations 766 Classical Equations and Boundary-Value Problems 771 Heat Equation 777 Wave Equation 780 Laplace's Equation 784 Nonhomogeneous Equations and Boundary Conditions 788 Use of Generalized Fourier Series 791 Boundary-Value Problems Involving Fourier Series in Two Variables 795 Summary 798 Chapter 12 Review Exercises BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS Problems Involving Laplace's Equation in Polar Coordinates Problems in Polar and Cylindrical Coordinates: Bessel Functions Problems in Spherical Coordinates: Legendre Polynomials 813 Summary 815 Chapter 13 Review Exercises INTEGRAL TRANSFORM METHOD Error Function Applications of the Laplace Transform 821

x Contents 14.3 Fourier Integral 828 14.4 Fourier Transforms 834 Summary 841 Chapter 14 Review Exercises 842 Part V NUMERICAL ANALYSIS 843 15 NUMERICAL METHODS 845 15.1 Newton's Method 846 15.

7 x Contents 14.3 Fourier Integral Fourier Transforms 834 Summary 841 Chapter 14 Review Exercises 842 Part V NUMERICAL ANALYSIS NUMERICAL METHODS Newton's Method Approximate Integration Direction Fields The Euler Methods The Three-Term Taylor Method The Runge-Kutta Method Multistep Methods, Errors Higher-Order Equations and Systems Second-Order Boundary-Value Problems Numerical Methods for Partial Differential Equations: Elliptic Equations Numerical Methods for Partial Differential Equations: Parabolic Equations Numerical Methods for Partial Differential Equations: Hyperbolic Equations Approximation of Eigenvalues 917 Summary 925 Chapter 15 Review Exercises 926 Part VI COMPLEX ANALYSIS FUNCTIONS OF A COMPLEX VARIABLE Complex Numbers Polar Form of Complex Numbers; Powers and Roots Set of Points in the Complex Plane Functions of a Complex Variable; Analyticity Cauchy-Riemann Equations Exponential and Logarithmic Functions Exponential Function Logarithmic Function Trigonometrie and Hyperbolic Functions Inverse Trigonometrie and Hyperbolic Functions 971 Summary 974 Chapter 16 Review Exercises 975

Contents xi 17 INTEGRATION IN THE COMPLEX PLANE 977 17.1 Contour Integrals 978 17.2 Cauchy-Goursat Theorem 984 17.3 Independence of Path 990 17.

8 Contents xi 17 INTEGRATION IN THE COMPLEX PLANE Contour Integrals Cauchy-Goursat Theorem Independence of Path Cauchy's Integral Formula 996 Summary 1003 Chapter 17 Review Exercises SERIES AND RESIDUES Sequences and Series Taylor Series Laurent Series Zeros and Poles Residues and Residue Theorem Evaluation of Real Integrals 1039 Summary 1047 Chapter 18 Review Exercises CONFORMAL MAPPINGS AND APPLICATIONS Complex Functions as Mappings Conformal Mapping and the Dirichlet Problem Linear Fractional Transformations Schwarz-Christoffel Transformations 1073 [O] 19.5 Poisson Integral Formulas Applications 1086 Summary 1095 Chapter 19 Review Exercises 1096 APPENDICES A-1 Appendix I Gamma Function A-3 Appendix II Table of Laplace Transforms A-6 Appendix III Conformal Mappings A-9 Appendix IV BASIC Programs for Numerical Methods in Chapter 15 A-20 Answers to Odd-Numbered Problems A-29 Index A-93

ADVANCED ENGINEERING MATHEMATICS MATLAB

ADVANCED ENGINEERING MATHEMATICS WITH MATLAB THIRD EDITION Dean G. Duffy Contents Dedication Contents Acknowledgments Author Introduction List of Definitions Chapter 1: Complex Variables 1.1 Complex Numbers