A Contemporary Approach to Teaching Differential Equations
Applied Differential Equations: An Introduction presents a contemporary treatment of ordinary differential equations (ODEs) and an introduction to partial differential equations (PDEs), including their applications in engineering and the sciences. Designed for a two-semester undergraduate course, the text offers a true alternative to books published for past generations of students. It enables students majoring in a range of fields to obtain a solid foundation in differential equations.
The text covers traditional material, along with novel approaches to mathematical modeling that harness the capabilities of numerical algorithms and popular computer software packages. It contains practical techniques for solving the equations as well as corresponding codes for numerical solvers. Many examples and exercises help students master effective solution techniques, including reliable numerical approximations.
This book describes differential equations in the context of applications and presents the main techniques needed for modeling and systems analysis. It teaches students how to formulate a mathematical model, solve differential equations analytically and numerically, analyze them qualitatively, and interpret the results.
Table of Contents
Equations with Homogeneous Coefficients
Exact Differential Equations
First-Order Linear Differential Equations
Equations Reducible to first Order
Existence and Uniqueness
Review Questions for Chapter 1
Applications of First Order ODE
Applications in Mathematics
Curves of Pursuit
Applications in Physics
Review Questions for Chapter 2
Mathematical Modeling and Numerical Methods
The Runge-Kutta Methods
Error Analysis and Stability
Review Questions for Chapter 3
Second and Higher Order Linear Equations
Linear Independence and Wronskians
The Fundamental Set of Solutions
Equations with Constant Coefficients
Repeated Roots. Reduction of Order
Variation of Parameters
Review Questions for Chapter 4
The Laplace Transform
Properties of the Laplace Transform
Discontinuous and Impulse Functions
The Inverse Laplace Transform
Applications to Homogenous Equations
Applications to Non-homogenous Equations
Review Questions for Chapter 5
Series of Solutions
Review of Power Series
Power Solutions about an Ordinary Point
Series Solutions Near a Regular Singular Point
Equations of Hypergeometric Type
Review Questions for Chapter 6
Applications of Higher Order Differential Equations
Boundary Value Problems
Some Numerical Methods
Dynamics of Rotational Motion
Modeling: Forced Oscillations
Modeling of Electric Circuits
Some Variational Problems
Review Questions for Chapter 7
Appendix: Software Packages
Answers to Problems
Vladimir A. Dobrushkin is an associate of Brown University.
"The author covers traditional material along with modern approaches for analyzing, solving, and visualizing ODEs. The topics are accompanied by mathematical software codes for some of the most popular packages … . The text contains a large number of examples from different areas … . It also includes advanced material for students who want to obtain a deeper knowledge on this subject. … the textbook contains a great number of exercises. In addition, each chapter ends with summary and review questions, making the text well-suited for self-study as well."
—Zentralblatt MATH 1326
"Two notable aspects of the book are its comprehensiveness and tight integration with applications. … There is much to like about this book — lucid writing, clear development of the basic ideas, and a very large number of exercises with a good range of difficulty. … a very attractive text."
—MAA Reviews, September 2015
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