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"A reader with a strong background in mathematics, at least two semesters of calculus, and interest in the social sciences will find the book helpful in learning how this area of mathematics can be used in different applications."

                                                                                                   —
S.L. Sullivan, Catawba College


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.

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Series Editor's Introduction
 
Acknowledgments
 
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 Versus 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  
An Example From Sociology  
Numerical Methods Used to Solve Differential Equations  
Summary  
Chapter 2 Appendix  
 
3. Systems of First-Order Differential Equations
The Predator-Prey Model  
The Phase Diagram  
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  
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  
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  
Summarizing the Stability Criteria  
 
7. Stability Analyses of Nonlinear Differential Equation Systems
The Jacobian  
Summary  
 
8. Frontiers of Exploration
 
References
 
Index
 
About the Author

"A reader with a strong background in mathematics, at least two semesters of calculus, and interest in the social sciences will find the book helpful in learning how this area of mathematics can be used in different applications."

S.L. Sullivan, Catawba College

S.L. Sullivan
Catawba College
CHOICE
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.


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