Elementary Differential Equations With Boundary Value Problems

by Edwards, C. Henry; Penney, David E.

• ISBN13: 9780136006138
• ISBN10: 0136006132
• Edition: 6th
• Format: Hardcover
• Copyright: 2007-11-30
• Publisher: Pearson

Summary

The Sixth Edition of this acclaimed differential equations book remains the same classic volume it's always been, but has been polished and sharpened to serve readers even more effectively. Offers precise and clear-cut statements of fundamental existence and uniqueness theorems to allow understanding of their role in this subject. Features a strong numerical approach that emphasizes that the effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques. Inserts new graphics and text where needed for improved accessibility. A useful reference for readers who need to brush up on differential equations.

Author Biography

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia's honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution's highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran's Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee's research team's primary focus was on the active transport of sodium ions by biological membranes. Penney's primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

Table of Contents

Preface

First-Order Differential Equations

Differential Equations and Mathematical Models

Integrals as General and Particular Solutions

Slope Fields and Solution Curves

Separable Equations and Applications

Linear First-Order Equations

Substitution Methods and Exact Equations

Population Models

Acceleration-Velocity Models

Linear Equations of Higher Order

Introduction: Second-Order Linear Equations

General Solutions of Linear Equations

Homogeneous Equations with Constant Coefficients

Mechanical Vibrations

Nonhomogeneous Equations and Undetermined Coefficients

Forced Oscillations and Resonance

Electrical Circuits

Endpoint Problems and Eigenvalues

Power Series Methods

Introduction and Review of Power Series

Series Solutions Near Ordinary Points

Regular Singular Points

Method of Frobenius: The Exceptional Cases

Bessel's Equation

Applications of Bessel Functions

LaplaceTransform Methods

Laplace Transforms and Inverse Transforms

Transformation of Initial Value Problems

Translation and Partial Fractions

Derivatives, Integrals, and Products of Transforms

Periodic and Piecewise Continuous Input Functions

Impulses and Delta Functions

Linear Systems of Differential Equations

First-Order Systems and Applications

The Method of Elimination

Matrices and Linear Systems

The Eigenvalue Method for Homogeneous Systems

Second-Order Systems and Mechanical Applications

Multiple Eigenvalue Solutions

Matrix Exponentials and Linear Systems

Nonhomogeneous Linear Systems

Numerical Methods

Numerical Approximation: Euler's Method

A Closer Look at the Euler Method

The Runge-Kutta Method

Numerical Methods for Systems

Nonlinear Systems and Phenomena

Equilibrium Solutions and Stability

Stability and the Phase Plane

Linear and Almost Linear Systems

Ecological Models: Predators and Competitors

Nonlinear Mechanical Systems

Chaos in Dynamical Systems

Fourier Series Methods

Periodic Functions and Trigonometric Series

General Fourier Series and Convergence

Fourier Sine and Cosine Series

Applications of Fourier Series

Heat Conduction and Separation of Variables

Vibrating Strings and the One-Dimensional Wave Equation

Steady-State Temperature and Laplace's Equation

Eigenvalues and Boundary Value Problems

Sturm-Liouville Problems and Eigenfunction Expansions

Applications of Eigenfunction Series

Steady Periodic Solutions and Natural Frequencies

Cylindrical Coordinate Problems

Higher-Dimensional Phenomena

References for Further Study

Existence and Uniqueness of Solutions

Answers to Selected Problems

Index I-1

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