Elementary Differential Equations with Boundary Value Problems

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

1. 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
2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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

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