A Course in Ordinary Differential Equations deserves to be on the MAA’s Basic Library List. The book is divided into eight chapters and an appendix. The eight chapters are Traditional First Order Differential Equations, Geometrical and Numerical Methods for First Order Equations, Elements of Higher Order Linear Equations, Techniques of Higher Order Linear Equations, Fundamentals of Systems of Differential Equations, Techniques of Systems of Differential Equations, Laplace Transforms and Series Methods. The Appendix is divided into four parts. The first part is an introduction to Matlab, Maple and Mathematica. The second and third parts, respectively, review graphing factored polynomials and some selected topics from Linear Algebra. The last part includes answers to selected exercises.
The first chapter starts with the usual basic terminology of Differential Equations. For each definition, the authors provide examples and explanations to make all the concepts clear. This process is continued throughout the book. Thus each chapter has sufficient examples and easy-to-follow explanations. The authors did not go into detailed mathematical theories or proving long theorems. Nor did they provide large number of examples, which could often overwhelm the students. Instead, the authors always kept their audience — undergraduate students who are not yet matured in Mathematics — in mind, and kept everything in context.
In that sense, the book with its layout, is very student-friendly — it is easy to read and understand; every chapter and explanations flow smoothly and coherently. Students would not be scared away by the book and each student would find something interesting in the book.
Mathematically inclined students would enjoy solving the varied types of problem exercises that are included at the end of each section. A great quality of this book is that it introduces students to solving problems in Mathematica, Maple and Matlab. From the very beginning of the book, the authors provide examples and computer codes written for all the three above mentioned scientific software packages. At the end of each chapter, projects and additional problems related to the chapter are also provided. Whenever possible, authors have also included application examples. So, non-mathematically inclined students from other sciences and engineering fields would also enjoy the book.
The book has some minor typos in the chapters and exercise problems. It is just a personal opinion of the reviewer that the introduction to the autonomous second order equations in chapter two should have been done after introducing higher orders in chapter three. Also instead of devoting a whole section to Essential Topics From Linear Algebra in chapter three, this material could well have been included in the Appendices.
The book primarily focuses on linear differential equations, though some nonlinear differential equations are included. Overall, the reviewer would recommend this book highly for undergraduate introductory differential equation courses.
Srabasti Dutta received her PhD from SUNY-Stony Brook and is currently an Assistant Professor in College of Saint Elizabeth. She can be reached at firstname.lastname@example.org.
TRADITIONAL FIRST ORDER DIFFERENTIAL EQUATIONS
Some Basic Terminology
Separable Differential Equations
Some Physical Problems arising as Separable Equations
GEOMETRICAL & NUMERICAL METHODS FOR FIRST ORDER EQUATIONS
Direction Fields - the Geometry of Differential Equations
Existence and Uniqueness for First Order Equations
First Order Autonomous Equations - Geometrical Insight
Population Modeling: An Application of Autonomous Equations
Numerical Approximation with the Euler Method
Numerical Approximation with the Runge-Kutta Method
An Introduction to Autonomous Second Order Equations
ELEMENTS OF HIGHER ORDER LINEAR EQUATIONS
Essential Topics from Linear Algebra
Reduction of Order - The Case n=2
Numerical Considerations for nth Order Equations
TECHNIQUES OF HIGHER ORDER LINEAR EQUATIONS
Homogeneous Equations with Constant Coefficients
A Mass on a Spring
Cauchy-Euler (Equidimensional) Equation
The Method of Undetermined Coefficients via Tables
The Method of Undetermined Coefficients via the Annihilator Method
Variation of Parameters
FUNDAMENTALS OF SYSTEMS OF DIFFERENTIAL EQUATIONS
Systems of Two Equations - Motivational Examples
Linear Transformations and the Fundamental Subspaces
Eigenvalues and Eigenvectors
TECHNIQUES OF SYSTEMS OF DIFFERENTIAL EQUATIONS
A General Method, Part I: Solving Systems with Real, Distinct Eigenvalues
A General Method, Part II: Solving Systems with Repeated Real or Complex Eigenvalues
Solving Linear Homogeneous and Nonhomogeneous Systems of Equations
Nonlinear Equations and Phase Plane Analysis
Fundamentals of the Laplace Transform
Properties of the Laplace Transforms
Step Functions, Translated Functions, and Periodic Functions
The Inverse Laplace Transform
Laplace Transform Solution of Linear Differential Equations
Solving Linear Systems using Laplace Transforms
Power Series Representations of Functions
The Power Series Method
Ordinary and Singular Points
The Method of Frobenius
Appendix A: An Introduction to MATLAB, Maple, and Mathematica
Appendix B: Graphing Factored Polynomials
Appendix C: Selected Topics from Linear Algebra
Appendix D: Answers to Selected Exercises