"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.
Author(s): Courtney Brown
Series: Quantitative Applications in the Social Sciences
Publisher: Sage
Year: 2007
Language: English
Pages: 121
Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Series Editor's Introduction......Page 9
Acknowledgments ......Page 11
1. Dynamic Models and Social Change ......Page 12
Theoretical Reasons for Using Differential Equations in the Social Sciences ......Page 14
An Example ......Page 16
The Use of Differential Equations in the Natural and Physical Sciences ......Page 18
Deterministic Versus Probabilistic Differential Equation Models ......Page 19
What Is a Differential Equation? ......Page 22
What This Book Is and Is Not ......Page 26
2. First-Order Differential Equations ......Page 28
Solving First-Order Differential Equations Using Separation of Variables ......Page 29
Exponential Growth ......Page 31
Exponential Decay ......Page 33
Learning Curves and Noninteractive Diffusion ......Page 34
Logistic Curve ......Page 36
An Example From Sociology ......Page 40
Numerical Methods Used to Solve Differential Equations ......Page 41
Euler?ˉs Method ......Page 42
Heun?ˉs Method ......Page 45
The Fourth-Order Runge-Kutta Method ......Page 46
Summary ......Page 47
Appendix ......Page 48
3. Systems of First-Order Differential Equations ......Page 51
The Predator-Prey Model ......Page 52
The Phase Diagram ......Page 55
Equilibria Within Phase Diagrams ......Page 57
Vector Field and Direction Field Diagrams ......Page 59
The Equilibrium Marsh and Flow Diagrams ......Page 64
Summary ......Page 66
Appendix ......Page 67
4. Some Classic Social Science Examples of First-Order Systems ......Page 70
Richardson?ˉs Arms Race Model ......Page 71
Lanchester?ˉs Combat Models ......Page 76
Scenario One ......Page 77
Rapoport?ˉs Production and Exchange Model ......Page 78
Summary ......Page 80
5. Transforming Second-Order and Nonautonomous Differential Equations Into Systems of First-Order Differential Equations ......Page 81
Second- and Higher-Order Differential Equations ......Page 83
An Example ......Page 84
Nonautonomous Differential Equations ......Page 85
6. Stability Analyses of Linear Differential Equation Systems ......Page 86
A Motivating Example of How Stability Can Dramatically Change in One System ......Page 87
Scalar Methods ......Page 88
Matrix Methods ......Page 92
Unstable Nodes ......Page 96
Stable Nodes ......Page 97
Saddle Points ......Page 98
Unstable Spirals ......Page 99
Stable Spirals ......Page 100
Ellipses ......Page 101
Summarizing the Stability Criteria ......Page 102
The Jacobian ......Page 104
An Example ......Page 107
8. Frontiers of Exploration ......Page 108
References ......Page 111
Index ......Page 113
About the Author ......Page 117
