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Published in 2007 by Sage Publications (Thousand Oaks, CA) as Volume 150 in the Quantitative Applications in the Social Sciences series.

Differential Equations

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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
    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
    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
  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
  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
  8. Frontiers of Exploration
    About the Author