Description
Book SynopsisC. 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
Table of ContentsTable of Contents
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First-Order Differential Equations
- 1.1 Differential Equations and Mathematical Models
- 1.2 Integrals as General and Particular Solutions
- 1.3 Slope Fields and Solution Curves
- 1.4 Separable Equations and Applications
- 1.5 Linear First-Order Equations
- 1.6 Substitution Methods and Exact Equations
- 1.7 Population Models
- 1.8 Acceleration-Velocity Models
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Linear Equations of Higher Order
- 2.1 Introduction: Second-Order Linear Equations
- 2.2 General Solutions of Linear Equations
- 2.3 Homogeneous Equations with Constant Coefficients
- 2.4 Mechanical Vibrations
- 2.5 Nonhomogeneous Equations and Undetermined Coefficients
- 2.6 Forced Oscillations and Resonance
- 2.7 Electrical Circuits
- 2.8 Endpoint Problems and Eigenvalues
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Power Series Methods
- 3.1 Introduction and Review of Power Series
- 3.2 Series Solutions Near Ordinary Points
- 3.3 Regular Singular Points
- 3.4 Method of Frobenius: The Exceptional Cases
- 3.5 Bessel's Equation
- 3.6 Applications of Bessel Functions
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Laplace Transform Methods
- 4.1 Laplace Transforms and Inverse Transforms
- 4.2 Transformation of Initial Value Problems
- 4.3 Translation and Partial Fractions
- 4.4 Derivatives, Integrals, and Products of Transforms
- 4.5 Periodic and Piecewise Continuous Input Functions
- 4.6 Impulses and Delta Functions
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Linear Systems of Differential Equations
- 5.1 First-Order Systems and Applications
- 5.2 The Method of Elimination
- 5.3 Matrices and Linear Systems
- 5.4 The Eigenvalue Method for Homogeneous Systems
- 5.5 Second-Order Systems and Mechanical Applications
- 5.6 Multiple Eigenvalue Solutions
- 5.7 Matrix Exponentials and Linear Systems
- 5.8 Nonhomogeneous Linear Systems
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Numerical Methods
- 6.1 Numerical Approximation: Euler's Method
- 6.2 A Closer Look at the Euler Method
- 6.3 The Runge-Kutta Method
- 6.4 Numerical Methods for Systems
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Nonlinear Systems and Phenomena
- 7.1 Equilibrium Solutions and Stability
- 7.2 Stability and the Phase Plane
- 7.3 Linear and Almost Linear Systems
- 7.4 Ecological Models: Predators and Competitors
- 7.5 Nonlinear Mechanical Systems
- 7.6 Chaos in Dynamical Systems
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Fourier Series Methods
- 8.1 Periodic Functions and Trigonometric Series
- 8.2 General Fourier Series and Convergence
- 8.3 Fourier Sine and Cosine Series
- 8.4 Applications of Fourier Series
- 8.5 Heat Conduction and Separation of Variables
- 8.6 Vibrating Strings and the One-Dimensional Wave Equation
- 8.7 Steady-State Temperature and Laplace's Equation
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Eigenvalues and Boundary Value Problems
- 9.1 Sturm-Liouville Problems and Eigenfunction Expansions
- 9.2 Applications of Eigenfunction Series
- 9.3 Steady Periodic Solutions and Natural Frequencies
- 9.4 Cylindrical Coordinate Problems
- 9.5 Higher-Dimensional Phenomena
References for Further Study Appendix: Existence and Uniqueness of Solutions Answers to Selected Problems Index
