Elementary Differential Equations

with

Boundary Value Problems T H I R D E D I T I O N

C. H. Edwards, jr. • David E. Penney The University of Georgia

PRENTICE-HALL, Englewood Cliffs, New Jersey 07632

Contents

Preface XI

1 First Order Differential Equations 1

1.1 Introduction 1.2 Solution by Direct Integration 1.3 Existence and Uniqueness of Solutions 1.4 Separable Equations and Applications 1.5 Linear First Order Equations 1.6 Substitution Methods 1.7 Exact Equations and Integrating Factors

"1.8 Population Models *1.9 Motion with Variable Acceleration

Summary and a Look Ahead Review Problems

2 11 19 32 46 57 67 76 84 98 101

Linear Equations of Higher Order 102

2.1 Introduction 103 2.2 General Solutions of Linear Equations 115 2.3 Homogeneous Equations with Constant Coefncients 126 2.4 Mechanical Vibrations 136 2.5 Nonhomogeneous Equations and the Method of Undetermined

Coefficients 149 2.6 Reduction of Order and Euler-Cauchy Equations 159 2.7 Variation of Parameters 171

*2.8 Forced Oscillations and Resonance 179 *2.9 Electrical Circuits 192

*2.10 Endpoint Problems and Eigenvalues 200

VII

Power Series Solutions of Linear Equations 215

3.1 Introduction and Review of Power Series 3.2 Series Solutions Near Ordinary Points 3.3 Regulär Singular Points

*3.4 Method of Frobenius: The Exceptional Cases 3.5 Bessel's Equation

*3.6 Applications of Bessel Functions *3.7 Appendix on Infinite Series and the Atom

216 230 240 254 267 277 284

4 The Laplace Transform 290

4.1 Laplace Transforms and Inverse Transforms 4.2 Transformation of Initial Value Problems 4.3 Translation and Partial Fractions 4.4 Derivatives, Integrals, and Products of Transforms

"4.5 Periodic and Piecewise Continuous Forcing Functions "4.6 Impulses and Delta Functions

Table of Laplace Transforms

291 302 312 320 328 341 353

Linear Systems of Differential Equations 354

5.1 Introduction to Systems 5.2 The Method of Elimination 5.3 Linear Systems and Matrices 5.4 The Eigenvalue Method for Homogeneous Systems

*5.5 Second Order Systems and Mechanical Applications 5.6 Multiple Eigenvalue Solutions 5.7 Nonhomogeneous Linear Systems

*5.8 Matrix Exponentials and Linear Systems

355 366 375 396 410 425 441 451

6 Numerical Methods 459

6.1 Introduction: Euler's Method 6.2 A Closer Look at the Euler Method, and Improvements 6.3 The Runge-Kutta Method 6.4 Systems of Differential Equations

460 468 479 488

7 Nonlinear Differential Equations and Systems 504

7.1 Introduction to Stability 7.2 Stability and the Phase Plane 7.3 Linear and Almost Linear Systems 7.4 Ecological Applications: Predators and Competitors 7.5 Nonlinear Mechanical Systems

s7.6 Chaos and Bifurcation

505 511 522 535 550 564

8 Fourier Series and Separation of Variables 582

8.1 Periodic Functions and Trigonometrie Series 583 8.2 General Fourier Series and Convergence 593 8.3 Even-Odd Functions and Termwise Differentiation 601

*ß.4 Applications of Fourier Series 612 8.5 Heat Conduction and Separation of Variables 619 8.6 Vibrating Strings and the One-Dimensional Wave Equation 631 8.7 Steady-State Temperature anc' Laplace's Equation 644

VIII Contents

Eigenvalues and Boundary Value Problems 658

9.1 Sturm-Liouville Problems and Eigenfunction Expansions 9.2 Applications of Eigenfunction Series *9«3 Steady Periodic Solutions and Natural Frequencies 9-4 Applications of Bessel Functions 9.5 Nuclear Reactors and Other Applications

659 670 681 689 704

References for Further Study

Appendix

Answers

Index

719

722

738

765

Contents IX