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
 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
 FirstOrder Differential Equations
Analytical Solutions to Linear FirstOrder Differential Equations
Solving FirstOrder Differential Equations Using Separation of
Variables
(i) Exponential Growth
(ii) Exponential Decay
(iii) Learning Curves and NonInteractive 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 Fourthorder RungeKutta Method
Summary
Chapter 2 Appendix
 Systems of FirstOrder Differential Equations
The PredatorPrey Model
The Phase Diagram
Equilibria Within Phase Diagrams
The PredatorPrey Model
Vector Field and Direction Field Diagrams
The Equilibrium Marsh and Flow Diagrams
Summary
Chapter 3 Appendix
 Some Classic Social Science Examples of FirstOrder
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
 Transforming SecondOrder and Nonautonomous Differential
Equations into Systems of FirstOrder Differential Equations
Second and HigherOrder Differential Equations
An Example
Nonautonomous Differential Equations
Summary
 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
 Stability Analyses of Nonlinear Differential Equation
Systems
The Jacobian
An Example
Summary
 Frontiers of Exploration
References
Index
About the Author
