ADVANCED ENGINEERING MATHEMATICS

DENNIS G. ZILL Loyola Marymount University

MICHAEL R. CULLEN Loyola Marymount University

O I 73 PWS-KENT ^ PUBLISHING COMPANY

E 9 U Boston

CONTENTS

Preface xiii

Parti ORDINARY DIFFERENTIAL EQUATIONS

1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 3

1.1 Basic Definitions and Terminology 4 1.2 Differential Equation of a Family of Curves 13 1.3 Mathematical Models 17

Summary 32 Chapter 1 Review Exercises 32

2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34

2.1 Preliminary Theory 35 2.2 Separable Variables 39 2.3 Homogeneous Equations 46 2.4 Exact Equations 52 2.5 Linear Equations 58

[O] 2.6 Equations of Bernoulli, Ricatti, and Clairaut 66 [O] 2.7 Substitutions 70 [O] 2.8 Picard's Method 73

2.9 Orthogonal Trajectories 76 2.10 Applications of Linear Equations 81 2.11 Applications of Nonlinear Equations 92

Summary 101 Chapter 2 Review Exercises 102

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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

3.2 Constructing a Second Solution from a Known Solution 126 3.3 Homogeneous Linear Equations with Constant Coefficients 131 3.4 Undetermined Coefficients 139 3.5 Differential Operators and Undetermined Coefficients

Revisited 148 3.5.1 Differential Operators 148

[O] 3.5.2 An Alternative Approach to Undetermined Coefficients 152

3.6 Variation of Parameters 157 3.7 Systems of Linear Differential Equations with Constant

Coefficients 164 3.8 Simple Harmonie Motion 172 3.9 Damped Motion 180 3.10 Forced Motion 189 3.11 Electric Circuits and Other Analogous Systems 198

Summary 204 Chapter 3 Review Exercises 206

4 LAPLACE TRANSFORM 208 4.1 Laplace Transform 209 4.2 Inverse Transform 217 4.3 Operational Properties 224

4.3.1 Translation Theorems and Derivatives of a Transform 224 4.3.2 Transforms of Derivatives and Integrals 233 4.3.3 Transform of a Periodic Function 237

4.4 Applications 242 4.5 Dirac Delta Function 255 4.6 Systems of Differential Equations 260

Summary 267 Chapter 4 Review Exercises 268

5 DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 270

5.1 Cauchy-Euler Equation 271 5.2 Power Series Solutions 278 5.3 Solutions About Singular Points 290 5.4 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 6.1 Vectors in the Plane 324 6.2 Vectors in Space 332 6.3 The Dot Product 339 6.4 The Cross Product 348 6.5 Lines and Planes in 3-Space 356 6.6 Vector Spaces 365

Summary 373 Chapter 6 Review Exercises 374

7 MATRICES 376

7.1 Matrix Algebra 377 7.2 Systems of Linear Algebraic Equations 387 7.3 Determinants 400 7.4 Properties of Determinants 406 7.5 Inverse of a Matrix 414

7.5.1 Finding the Inverse 414 7.5.2 Using the Inverse to Solve Systems 421

7.6 Cramer's Ruie 426 7.7 The Eigenvalue Problem 430 7.8 Orthogonal Matrices 437 7.9 Diagonalization 444 7.10 Cryptography 454 7.11 An Error-Correcting Code 458 7.12 Method of Least Squares 465

Summary 469 Chapter 7 Review Exercises 469

8 VECTOR CALCULUS 473

8.1 Vector Functions 474 8.2 Motion on a Curve; Velocity and Acceleration 482 8.3 Curvature; Components of Acceleration 488 8.4 Functions of Several Variables; Chain Ruie 494 8.5 The Directional Derivative 502

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8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17

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 617

9 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 619

9.1 Systems in Normal Form 620 9.2 Matrix Functions 625 9.3 Preliminary Theory 629 9.4 Homogeneous Linear Systems 641

9.4.1 Distinct Real Eigenvalues 641 9.4.2 Complex Eigenvalues 645 9.4.3 Repeated Eigenvalues 649

9.5 Solution by Diagonalization 656 9.6 Nonhomogeneous Linear Systems 658

9.6.1 Undetermined Coefficients 658 9.6.2 Variation of Parameters 661 9.6.3 Diagonalization 664

[O] 9.7 Matrix Exponential 667 Summary 670 Chapter 9 Review Exercises 671

10 PLANE AUTONOMOUS SYSTEMS AND STABILITY 673

10.1 Autonomous Systems, Critical Points, and Periodic Solutions 674 10.2 Stability and Linear Systems 680 10.3 Linearization and Local Stability 688 10.4 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 11.3 Fourier Cosine and Sine Series 737 11.4 Sturm-Liouville Problem 746 11.5 Bessel and Legendre Series 755

11.5.1 Fourier-Bessel Series 755 11.5.2 Fourier-Legendre Series 759 Summary 762 Chapter 11 Review Exercises 763

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 799

13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 801

13.1 Problems Involving Laplace's Equation in Polar Coordinates 802 13.2 Problems in Polar and Cylindrical Coordinates: Bessel

Functions 807 13.3 Problems in Spherical Coordinates: Legendre Polynomials 813

Summary 815 Chapter 13 Review Exercises 816

14 INTEGRAL TRANSFORM METHOD 818 14.1 Error Function 819 14.2 Applications of the Laplace Transform 821

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12.1 12.2 12.3 12.4 12.5 12.6 12.7

[O] 12.8

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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 15.2 Approximate Integration 854 15.3 Direction Fields 866 15.4 The Euler Methods 871 15.5 The Three-Term Taylor Method 877 15.6 The Runge-Kutta Method 880 15.7 Multistep Methods, Errors 885 15.8 Higher-Order Equations and Systems 888 15.9 Second-Order Boundary-Value Problems 892 15.10 Numerical Methods for Partial Differential Equations: Elliptic

Equations 896 15.11 Numerical Methods for Partial Differential Equations: Parabolic

Equations 903 15.12 Numerical Methods for Partial Differential Equations: Hyperbolic

Equations 911 15.13 Approximation of Eigenvalues 917

Summary 925 Chapter 15 Review Exercises 926

Part VI COMPLEX ANALYSIS 929

16 FUNCTIONS OF A COMPLEX VARIABLE 931 16.1 Complex Numbers 932 16.2 Polar Form of Complex Numbers; Powers and Roots 936 16.3 Set of Points in the Complex Plane 942 16.4 Functions of a Complex Variable; Analyticity 945 16.5 Cauchy-Riemann Equations 952 16.6 Exponential and Logarithmic Functions 958

16.6.1 Exponential Function 958 16.6.2 Logarithmic Function 962

16.7 Trigonometrie and Hyperbolic Functions 967 16.8 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 17.4 Cauchy's Integral Formula 996

Summary 1003 Chapter 17 Review Exercises 1004

18 SERIES AND RESIDUES 1006 18.1 Sequences and Series 1007 18.2 Taylor Series 1013 18.3 Laurent Series 1020 18.4 Zeros and Poles 1028 18.5 Residues and Residue Theorem 1032 18.6 Evaluation of Real Integrals 1039

Summary 1047 Chapter 18 Review Exercises 1048

19 CONFORMAL MAPPINGS AND APPLICATIONS 1050

19.1 Complex Functions as Mappings 1051 19.2 Conformal Mapping and the Dirichlet Problem 1056 19.3 Linear Fractional Transformations 1066 19.4 Schwarz-Christoffel Transformations 1073

[O] 19.5 Poisson Integral Formulas 1080 19.6 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