Differential Equations

Size: px

Start display at page:

Download "Differential Equations"

Silvester Austin
6 years ago
Views:

1 Differential Equations Theory, Technique, and Practice George F. Simmons and Steven G. Krantz Higher Education Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

2 TABLE OF CONTENTS r. PREFACE. VIII CHAPTER 1 WHAT IS A DIFFERENTIAL EQUATION? The Nature of Solutions Separable Equations First-Order Linear Equations Exact Equations Orthogonal Trajectories and Families of Curves Homogeneous Equations Integrating Factors Reduction of Order Dependent Variable Missing Independent Variable Missing The Hanging Chain and Pursuit Curves The Hanging Chain Pursuit Curves Electrical Circuits Anatomy of an Application: The Design of a Dialysis Machine CHAPTER 2 SECOND-ORDER LINEAR EQUATIONS Second-Order Linear Equations with Constant Coefficients The Method of Undetermined Coefficients The Method of Variation of Parameters The Use of a Known Solution to Find Another Vibrations and Oscillations Undamped Simple Harmonic Motion Damped Vibrations Forced Vibrations A Few Remarks "about Electricity in

3 iv Table of Contents 2.6 Newton's Law of Gravitation and Kepler's Laws Kepler's Second Law Kepler's First Law Kepler's Third Law Higher Order Linear Equations, Coupled Harmonic Oscillators 93 Historical Note: Euler 99 Anatomy of an Application: Bessel Functions and the Vibrating Membrane CHAPTER 3 QUALITATIVE PROPERTIES AND THEORETICAL ASPECTS Review of Linear Algebra Vector Spaces The Concept of Linear Independence Bases Inner Product Spaces Linear Transformations and Matrices Eigenvalues and Eigenvectors A Bit of Theory Picard's Existence and Uniqueness Theorem The Form of a Differential Equation Picard's Iteration Technique Some Illustrative Examples Estimation of the Picard Iterates Oscillations and the Sturm Separation Theorem The Sturm Comparison Theorem 138 Anatomy of an Application-. The Green's Function CHAPTER4 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 4.1 Introduction and Review of Power Series Review of Power Series 4.2 Series Solutions of First-Order Differential Equations

4 Table of Contents v 4.3 Second-Order Linear Equations: Ordinary Points Regular Singular Points More on Regular Singular Points Gauss's Hypergeometric Equation 184 Historical Note: Gauss 189 Historical Note: Abel 190 Anatomy of an Application: Steady-State Temperature in a Ball CHAPTER 5 FOURIER SERIES: BASIC CONCEPTS Fourier Coefficients Some Remarks about Convergence Even and Odd Functions: Cosine and Sine Series Fourier Series on Arbitrary Intervals Orthogonal Functions 221 Historical Note: Riemann 225 Anatomy of an Application: Introduction to the Fourier Transform CHAPTER 6 PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS Introduction and Historical Remarks Eigenvalues, Eigenfunctions, and the Vibrating String Boundary Value Problems Derivation of the Wave Equation Solution of the Wave Equation The Heat Equation The Dirichlet Problem for a Disc The Poisson Integral Sturm-Liouville Problems 262 Historical Note: Fourier 267 Historical Note: Dirichlet 268 Anatomy of an Application: Some Ideas from Quantum Mechanics

5 VI Table of Contents CHAPTER LAPLACE TRANSFORMS Introduction Applications to Differential Equations Derivatives and Integrals of Laplace Transforms Convolutions Abel's Mechanical Problem The Unit Step and Impulse Functions Historical Note: Laplace Anatomy of an Application: Flow Initiated by an Impulsively Started Flat Plate CHAPTER 8 THE CALCULUS OF VARIATIONS Euler's Equation Isoperimetric Problems and the Like Lagrange Multipliers Integral Side Conditions Finite Side Conditions Historical Note: Newton Anatomy of an Application: Hamilton's Principle and its Implications CHAPTER 9 NUMERICAL METHODS The Method of Euler The Error Term An Improved Euler Method The Runge-Kutta Method Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations CHAPTER 1 0 SYSTEMS OF FIRST-ORDER EQUATIONS Linear Systems

6 Table of Contents vii 10.3 Homogeneous Linear Systems with Constant Coefficients Nonlinear Systems: Volterra's Predator-Prey Equations 389 Anatomy of an Application: Solution of Systems with Matrices and Exponentials CHAPTER 11 THE NONLINEAR THEORY Some Motivating Examples Specializing Down Types of Critical Points: Stability Critical Points and Stability for Linear Systems Stability by Liapunov's Direct Method Simple Critical Points of Nonlinear Systems Nonlinear Mechanics: Conservative Systems Periodic Solutions: The Poincare-Bendixson Theorem 444 Historical Note: Poincare 452 Anatomy of an Application: Mechanical Analysis of a Block on a Spring CHAPTER 12 DYNAMICAL SYSTEMS Flows Dynamical Systems Stable and Unstable Fixed Points Linear Dynamics in the Plane Some Ideas from Topology Open and Closed Sets The Idea of Connectedness Closed Curves in the Plane Planar Autonomous Systems Ingredients of the Proof of Poincare-Bendixson 480 Anatomy of an Application: Lagrange's, Equations BIBLIOGRAPHY 495 ANSWERS TO ODD-NUMBERED EXERCISES 497 INDEX ; 525

Differential Equations with Mathematica

Differential Equations with Mathematica THIRD EDITION Martha L. Abell James P. Braselton ELSEVIER ACADEMIC PRESS Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore