From the publisher's description of the book:

Differential Equations: A Modeling Approach introduces differential equations and differential equation modeling to students and researchers in the social sciences. The text explains the mathematics and theory of differential equations. Graphical methods of analysis are emphasized over formal proofs, making the text even more accessible for newcomers to the subject matter. This volume introduces the subject of ordinary differential equations — as well as systems of such equations — to the social science audience. Social science examples are used extensively, and readers are guided through the most elementary models to much more advanced specifications. Emphasis is placed on easily applied and broadly applicable numerical methods for solving differential equations, thereby avoiding complicated mathematical “tricks” that often do not even work with more interesting nonlinear models. Also, graphical methods of analysis are introduced that allow social scientists to rapidly access the power of sophisticated model specifications. This volume also describes in clear language how to evaluate the stability of a system of differential equations (linear or nonlinear) by using the system’s eigenvalues. The mixture of nonlinearity with dynamical systems is a virtual trademark for this author’s approach to modeling, and this theme comes through clearly throughout this volume. This volume’s clarity of exposition encourages social science students of mathematical modeling to begin working with differential equation models that address complex and sophisticated social theories.

Key Features:

The text is accessibly written, so that students with minimal mathematical training can understand all of the basic concepts and techniques presented.

The author uses social sciences examples to illustrate the relevance of differential equation modeling to readers.

Readers can use graphical methods to produce penetrating analysis of differential equation systems.

Linear and nonlinear model specifications are explained from a social science perspective. Most interesting differential equation models are nonlinear, and readers need to know how to specify and work with such models in the social sciences.

Table of Contents

1. Dynamic Models and Social Change
   Theoretical Reasons for Using Differential Equations in the Social Sciences
   An Example
   The Use of Differential Equations in the Natural and Physical Sciences
   Deterministic Vs. Probabilistic Differential Equation Models
   What Is a Differential Equation?
   What This Book Is and Is Not
2. First-Order Differential Equations
   Analytical Solutions to Linear First-Order Differential Equations
   Solving First-Order Differential Equations Using Separation of Variables
   (i) Exponential Growth
   (ii) Exponential Decay
   (iii) Learning Curves and Non-Interactive Diffusion
   (iv) Logistic Curve
   An Example from Sociology
   Numerical Methods Used to Solve Differential Equations
   (i) Euler’s Method
   (ii) Heun’s Method
   (iii) The Fourth-order Runge-Kutta Method
   Summary
   Chapter 2 Appendix
3. Systems of First-Order Differential Equations
   The Predator-Prey Model
   The Phase Diagram
   Equilibria Within Phase Diagrams
   The Predator-Prey Model
   Vector Field and Direction Field Diagrams
   The Equilibrium Marsh and Flow Diagrams
   Summary
   Chapter 3 Appendix
4. Some Classic Social Science Examples of First-Order Systems
   Richardson’s Arms Race Model
   Lanchester’s Combat Model
   (i) Scenario One
   (ii) Scenario Two
   (iii) Scenario Three
   Rapoport’s Production and Exchange Model
   Summary
5. Transforming Second-Order and Nonautonomous Differential Equations into Systems of First-Order Differential Equations
   Second- and Higher-Order Differential Equations
   An Example
   Nonautonomous Differential Equations
   Summary
6. Stability Analyses of Linear Differential Equation Systems
   A Motivating Example of How Stability Can Dramatically Change in One System
   Scalar Methods
   Matrix Methods
   Equilibrium Categories
   (i) Unstable Nodes
   (ii) Stable Nodes
   (iii) Saddle Points
   (iv) Unstable Spirals
   (v) Stable Spirals
   (vi) Ellipses
   Summarizing the Stability Criteria
7. Stability Analyses of Nonlinear Differential Equation Systems
   The Jacobian
   An Example
   Summary
8. Frontiers of Exploration
   References
   Index
   About the Author