1. Introduction
1.1 Historical Prologue
1.2 Definitions and Terminology
1.3 Solutions and Problems
1.4 A Nobel Prize Winning Application
2. The Initial Value Problem y’ = f (x, y); y (c) = d
2.1 Direction Fields
2.2 Fundamental Theorems
2.3 Solution of Simple First-Order Differential Equations
2.4 Numerical Solution
3. Applications of the Initial Value Problem y’ = f (x, y); y (c) = d
3.1 Calculus Revisited
3.2 Learning Theory Models
3.3 Population Models
3.4 Simple Epidemic Models
3.5 Falling Bodies
3.6 Mixture Problems
3.7 Curves of Pursuit
3.8 Chemical Reactions
3.9 Miscellaneous Exercises
4. N-th Order Linear Differential Equations
4.1 Basic Theory
4.2 Roots of Polynomials
4.3 Homogeneous Linear Equations with Constant Coefficients
4.4 Nonhomogeneous Linear Equations with Constant Coefficients
4.5 Initial Value Problems
5. The Laplace Transform Method
5.1 The Laplace Transform and Its Properties
5.2 Using the Laplace Transform and Its Inverse to Solve Initial Value Problems
5.3 Convolution and the Laplace Transform
5.4 The Unit Function and Time-Delay Function
5.5 Impulse Function
6. Applications of Linear Differential Equations with Constant Coefficients
6.1 Second-Order Differential Equations
6.2 Higher Order Differential Equations
7. Systems of First-Order Differential Equations
7.1 Properties of Systems of Differential Equations
7.2 Writing Systems as Equivalent First-Order Systems
8. Linear Systems of First-Order Differential Equations
8.1 Matrices and Vectors
8.2 Eigenvalues and Eigenvectors
8.3 Linear Systems with Constant Coefficients
9. Applications of Linear Systems with Constant Coefficients
9.1 Coupled Spring-Mass Systems
9.2 Pendulum Systems
9.3 The Path of an Electron
9.4 Mixture Problems
10. Applications of Systems of Equations
10.1 Richardson’s Arms Race Model
10.2 Phase-Plane Portraits
10.3 Modified Richardson’s Arms Race Models
10.4 Lanchester’s Combat Models
10.5 Models for Interacting Species
10.6 Epidemics
10.7 Pendulums
10.8 Duffing’s Equation (Nonlinear Spring-Mass Systems)
10.9 Van Der Pol’s Equation
10.10 Mixture Problems
10.11 The Restricted Three-Body Problem
11. Applications to Combinatorics
11.1 Power series
11.2 Set partitions
11.3 Trees
A. Appendix
A.1 Single-Step Methods
A.2 Multistep Methods
A.3 Predictor-Corrector Methods
Biography
Charles E. Roberts, Jr. (deceased) was Professor Emeritus in the Department of Mathematics and Computer Science at Indiana State University. He is remembered as a master teacher and prolific researcher. His book, A Mathematical Introduction to Proofs, is also published by CRC Press.
Miklós Bóna received his Ph.D in mathematics from the Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where, in 2010, he was inducted into the Academy of Distinguished Teaching Scholars. Professor Bóna has mentored numerous graduate and undergraduate students. He is the author of five books and more than 100 research articles, mostly focusing on enumerative and analytic combinatorics. His book, Combinatorics of Permutations, won a 2006 Outstanding Title Award from Choice, the journal of the American Library Association. He is also an Editor-in-Chief for the Electronic Journal of Combinatorics, and for two book series at CRC Press.
