—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.
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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|Theoretical Reasons for Using Differential Equations in the Social Sciences|
|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|
|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|
|Chapter 2 Appendix|
|The Predator-Prey Model|
|The Phase Diagram|
|Vector Field and Direction Field Diagrams|
|The Equilibrium Marsh and Flow Diagrams|
|Chapter 3 Appendix|
|Richardson's Arms Race Model|
|Lanchester's Combat Model|
|Rapoport's Production and Exchange Model|
|Second- and Higher-Order Differential Equations|
|Nonautonomous Differential Equations|
|A Motivating Example of How Stability Can Dramatically Change in One System|
|Summarizing the Stability Criteria|