Differential Equations with Boundary Value Problems

Size: px

Start display at page:

Download "Differential Equations with Boundary Value Problems"

Cameron Carmella Collins
5 years ago
Views:

1 Differential Equations with Boundary Value Problems John Polking Rice University Albert Boggess Texas A&M University David Arnold College of the Redwoods Pearson Education, Inc. Upper Saddle River, New Jersey 07458

2 Contents Preface ix 1 Introduction to Differential Equations Differential Equation Models The Derivative Integration 10 2 First-Order Equations Differential Equations and Solutions Solutions to Separable Equations Models of Motion Linear Equations Mixing Problems Exact Differential Equations Existence and Uniqueness of Solutions Dependence of Solutions on Initial Conditions Autonomous Equations and Stability 107 Project 2.10 The Daredevil Skydiver Modeling and Applications Modeling Population Growth Models and the Real World Personal Finance Electrical Circuits 152 Project 3.5 The Spruce Budworm 158, Project 3.6 Social Security, Now or Later Second-Order Equations Definitions and Examples Second-Order Equations and Systems Linear, Homogeneous Equations with Constant Coefficients Harmonic Motion 190

3 Vi Contents 4.5 Inhomogeneous Equations; the Method of Undetermined Coefficients Variation of Parameters Forced Harmonic Motion 215 Project 4.8 Nonlinear Oscillators The Laplace Transform The Definition of the Laplace Transform Basic Properties of the Laplace Transform The Inverse Laplace Transform Using the Laplace Transform to Solve Differential Equations Discontinuous Forcing Terms The Delta Function Convolutions Summary 298 Project 5.9 Forced Harmonic Oscillators Numerical Methods Euler's Method Runge-Kutta Methods Numerical Error Comparisons Practical Use of Solvers A Cautionary Tale 332 Project 6.6 Numerical Error Comparison Matrix Algebra Vectors and Matrices The Geometry of Systems of Linear Equations Solving Systems of Equations Properties of Solution Sets Subspaces Determinants An Introduction to Systems Definitions and Examples Geometric Interpretation of Solutions Qualitative Analysis Linear Systems 425 Project 8.5 Long-Term Behavior of Solutions 441

4 Contents vii 9.1 Overview of the Technique Planar Systems Phase Plane Portraits Higher Dimensional Systems The Exponential of a Matrix Qualitative Analysis of Linear Systems Higher-Order Linear Equations Inhomogeneous Linear Systems 528 Project 9.9 Phase Plane Portraits 538 Project 9.10 Oscillations of Linear Molecules Nonlinear Systems The Linearization of a Nonlinear System Long-Term Behavior of Solutions Invariant Sets and the Use of Nullclines Long-Term Behavior of Solutions to Planar Systems Conserved Quantities Nonlinear Mechanics The Method of Lyapunov Predator-Prey Systems 619 Project 10.9 Human Immune Response to Infectious Disease 631 Project Analysis of Competing Species Series Solutions to Differential Equations Review of Power Series Series Solutions Near Ordinary Points Legendre's Equation Types of Singular Points Euler's Equation Series Solutions Near Regular Singular Points Solutions in the Exceptional Cases Bessel's Equation and Bessel Functions Fourier Series Computation of Fourier Series Convergence of Fourier Series Fourier Cosine and Sine Series The Complex Form of a Fourier Series The Discrete Fourier Transform and the FFT 741

5 Viii Contents 1 3 Partial Differential Equations Derivation of the Heat Equation Separation of Variables for the Heat Equation The Wave Equation Laplace's Equation Laplace's Equation on a Disk Sturm Liouville Problems Orthogonality and Generalized Fourier Series Temperature in a Ball Legendre Polynomials Time Dependent PDEs in Higher Dimension Domains with Circular Symmetry Bessel Functions 821 Answers to Odd-Numbered Problems A-l Index l-l

CHAPTER 1 Introduction to Differential Equations 1 CHAPTER 2 First-Order Equations 29

Contents PREFACE xiii CHAPTER 1 Introduction to Differential Equations 1 1.1 Introduction to Differential Equations: Vocabulary... 2 Exercises 1.1 10 1.2 A Graphical Approach to Solutions: Slope Fields