DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS

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1 DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS Modern Methods and Applications 2nd Edition International Student Version James R. Brannan Clemson University William E. Boyce Rensselaer Polytechnic Institute WILEY John Wiley & Sons, Inc.

2 C O N T E N T S 1 Introduction Mathematical Models, Solutions, and Direction Fields Linear Equations: Method of Integrating Factors Numerical Approximations: Euler's Method Classification of Differential Equations 37 First Order Differential Equations Separable Equations Modeling with First Order Equations Differences Between Linear and Nonlinear Equations Autonomous Equations and Population Dynamics Exact Equations and Integrating Factors Accuracy of Numerical Methods Improved Euler and Runge-Kutta Methods P.1 Harvesting a Renewable Resource P.2 Designing a Drip Dispenser for a Hydrology Experiment P.3 A Mathematical Model of a Groundwater Contaminant Source P.4 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin 120 Systems of Two First Order Equations Systems of Two Linear Algebraic Equations Systems of Two First Order Linear Differential Equations Homogeneous Linear Systems with Constant Coefficients Complex Eigenvalues Repeated Eigenvalues A Brief Introduction to Nonlinear Systems Numerical Methods for Systems of First Order Equations P.1 Eigenvalue-Placement Design of a Satellite Attitude Control System P.2 Estimating Rate Constants for an Open Two-Compartment Model P.3 The Ray Theory of Wave Propagation P.4 A Blood-Brain Pharmacokinetic Model 211 Second Order Linear Equations Definitions and Examples Theory of Second Order Linear Homogeneous Equations Linear Homogeneous Equations with Constant Coefficients Mechanical and Electrical Vibrations Nonhomogeneous Equations; Method of Undetermined Coefficients Forced Vibrations, Frequency Response, and Resonance Variation of Parameters 285

3 xxii I Contents 4.P.1 A Vibration Insulation Problem P.2 Linearization of a Nonlinear Mechanical System P.3 A Spring-Mass Event Problem P.4 Uniformly Distributing Points on a Sphere P.5 Euler-Lagrange Equations 304 The Laplace Transform Definition of the Laplace Transform Properties of the Laplace Transform The Inverse Laplace Transform Solving Differential Equations with Laplace Transforms Discontinuous Functions and Periodic Functions Differential Equations with Discontinuous Forcing Functions Impulse Functions Convolution Integrals and Their Applications Linear Systems and Feedback Control P.1 An Electric Circuit Problem P.2 Effects of Pole Locations on Step Responses of Second Order Systems P.3 The Watt Governor, Feedback Control, and Stability Systems of First Order Linear Equations Definitions and Examples Basic Theory of First Order Linear Systems Homogeneous Linear Systems with Constant Coefficients Nondefective Matrices with Complex Eigenvalues Fundamental Matrices and the Exponential of a Matrix Nonhomogeneous Linear Systems Defective Matrices P.1 ACompartment Model of Heat Flow in a Rod P.2 Earthquakes and Tall Buildings P.3 Controlling a Spring-Mass System to Equilibrium 468 Nonlinear Differential Equations and Stability Autonomous Systems and Stability Almost Linear Systems Competing Species Predator-Prey Equations Periodic Solutions and Limit Cycles Chaos and Strange Attractors: The Lorenz Equations P.1 Modeling of Epidemics P.2 Harvesting in a Competitive Environment P.3 The Rossler System 540

4 Contents xxiii 8 Series Solutions of Second Order Linear Equations Review of Power Series Series Solutions Near an Ordinary Point, Part I Series Solutions Near an Ordinary Point, Part II Regular Singular Points Series Solutions Near a Regular Singular Point, Part I Series Solutions Near a Regular Singular Point, Part II Bessel's Equation P.1 Diffraction Through a Circular Aperture P.2 Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator P.3 Perturbation Methods 613 Partial Differential Equations and Fourier Series Two-Point Boundary Value Problems Fourier Series The Fourier Convergence Theorem Even and Odd Functions Separation of Variables; Heat Conduction in a Rod Other Heat Conduction Problems The Wave Equation: Vibrations of an Elastic String Laplace's Equation 684 Appendixes 9.A Derivation of the Heat Equation B Derivation of the Wave Equation P.1 Estimating the Diffusion Coefficient in the Heat Equation P.2 The Transmission Line Problem P.3 Solving Poisson's Equation by Finite Differences Boundary Value Problems and Sturm-Liouville Theory The Occurrence of Two-Point Boundary Value Problems Sturm-Liouville Boundary Value Problems Nonhomogeneous Boundary Value Problems Singular Sturm-Liouville Problems Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion Series of Orthogonal Functions: Mean Convergence P.1 Dynamic Behavior of a Hanging Cable P.2 Advection Dispersion: A Model for Solute Transport in Saturated Porous Media P.3 Fisher's Equation for Population Growth and Dispersion 782

5 xxiv Contents Matrices and Linear Algebra 787 A.1 Matrices 787 A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank' 796 A.3 Determinants and Inverses 813 A.4' The.Eigenvalue Problem 822' D Complex Variables 833 D ANSWERS TO SELECTED PROBLEMS 838 REFERENCES 944 PHOTO CREDITS 947 INDEX 948

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