Differential Equations and Linear Algebra

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

• ISBN13: 9780136054252
• ISBN10: 0136054250
• Edition: 3rd
• Format: Hardcover
• Copyright: 2008-10-10
• Publisher: Pearson

Summary

Acclaimed authors Edwards and Penney combine core topics in elementary differential equations with those concepts and methods of elementary linear algebra needed for a contemporary combined introduction to differential equations and linear algebra. Known for its real-world applications and its blend of algebraic and geometric approaches, this book discusses mathematical modeling of real-world phenomena, with a fresh new computational and qualitative flavor evident throughout in figures, examples, problems, and applications. First-Order Differential Equations; Mathematical Models and Numerical Methods; Linear Systems and Matrices; Vector Spaces; Higher-Order Linear Differential Equations; Eigenvalues and Eigenvectors; Linear Systems of Differential Equations; Matrix Exponential Methods; Nonlinear Systems and Phenomena; Laplace Transform Methods; Power Series Methods. For future math majors, engineers, or scientists that have taken two or three semesters of Calculus.

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

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

Mathematical Models and Numerical Methods

Population Models

Equilibrium Solutions and Stability

Accelerationndash;Velocity Models

Numerical Approximation: Euler's Method

A Closer Look at the Euler Method

The Rungendash;Kutta Method

Linear Systems and Matrices

Introduction to Linear Systems

Matrices and Gaussian Elimination

Reduced Row-Echelon Matrices

Matrix Operations

Inverses of Matrices

Determinants

Linear Equations and Curve Fitting

Vector Spaces

The Vector Space R3

The Vector Space Rn and Subspaces

Linear Combinations and Independence of Vectors

Bases and Dimension for Vector Spaces

Row and Column Spaces

Orthogonal Vectors in Rn

General Vector Spaces

Higher-Order Linear Differential Equations

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

Eigenvalues and Eigenvectors

Introduction to Eigenvalues

Diagonalization of Matrices

Applications Involving Powers of Matrices

Linear Systems of Differential Equations

First-Order Systems and Applications

Matrices and Linear Systems

The Eigenvalue Method for Linear Systems

Second-Order Systems and Mechanical Applications

Multiple Eigenvalue Solutions

Numerical Methods for Systems

Matrix Exponential Methods

Matrix Exponentials and Linear Systems

Nonhomogeneous Linear Systems

Spectral Decomposition Methods

Nonlinear Systems and Phenomena

Stability and the Phase Plane

Linear and Almost Linear Systems

Ecological Models: Predators and Competitors

Nonlinear Mechanical Systems

Laplace Transform 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

Power Series Methods

Introduction and Review of Power Series

Power Series Solutions

Frobenius Series Solutions

Existence and Uniqueness of Solutions

Theory of Determinants Answers to Selected Problems

Index

Application Modules The modules listed here follow the indicated sections in the text

Most provide computing projects that illustrate the corresponding text sections

Maple, Mathematica, and MATLAB versions of these investigations are included in the Applications Manual that accompanies this textbook

Computer-Generated Slope Fields and Solution Curves

The Logistic Equation

Indoor Temperature Oscillations

Computer Algebra Solutions

Logistic Modeling of Population Data

Rocket Propulsion

Implementing Euler's Method

Improved Euler Implementation

Runge-Kutta Implementation

Automated Row Operations

Automated Row Reduction

Automated Solution of Linear Systems

Plotting Seco

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