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Differential Equations: A Toolbox for Modeling the World

Kurt Bryan
Publication Date: 
Number of Pages: 
Electronic Book
[Reviewed by
Gilberto Gonzalez-Parra
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This textbook was written for a first course in ordinary differential equations for undergraduate students pursuing majors in mathematics, engineering, and science. The approach of the book is to use real-world phenomena to construct mathematical models based on differential equations in such a way that undergraduate students are motivated to learn the material and understand the usefulness of differential equations in a variety of problems.  

This book can also be useful for many non-specialist readers who may be interested in some real-world applications of differential equations. It includes explanations of the objectives and initial assumptions used in the models. The book starts with a stimulating real-world situation related to the Olympic 100-Meter Dash and presents a model that attempts to describe the position of a sprinter during the 100-meters. Examples like this help to motivate students to observe immediate applications of differential equations.

The author covers essential topics for a first course in ordinary differential equations, such as first-order linear equations, direction fields, second-order differential equations and Laplace transforms to solve differential equations.  In addition to these topics, the book includes some basic numerical methods to solve differential equations and linear and nonlinear systems of differential equations. Thus, the book covers essential aspects of the first course on differential equations plus additional material that is edifying for both students and general readers. Interestingly, the book also briefly covers deeper topics such as bifurcations, nondimensional rescaling and the matrix exponential, which serves as a rapid introduction to these areas.  The only aspect related to ordinary differential equations that this book does not include is series solutions.

In general, all the chapters include interesting real-world examples. Chapter 6 includes the linear model for the double spring-mass system. Chapter 7 includes a nonlinear mathematical model for epidemics that have been used to model a variety of diseases including the current COVID-19 pandemic.  Besides these, it covers essential tools to study nonlinear systems of differential equations such as linearization, phase portraits and the Hartman-Grobman Theorem. Finally, this book has at the end of each section and chapter a section devoted to exercises and proposed projects that allow students to better grasp the particular content of the chapter. These real-world problems are interesting and complement the ideas and material of the book. Instructors can assign projects in a judicious manner to enhance the learning process of the students.

Gilberto Gonzalez-Parra is an assistant professor of mathematics at New Mexico Tech.