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Applied Differential Equations with Boundary Value Problems

Vladimir A. Dobrushkin
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics
[Reviewed by
Bill Satzer
, on

This is a comprehensive treatment of ordinary differential equations that includes a short introduction to partial differential equations. It is designed for a two-semester course with second- and third-year undergraduates and has a strong orientation toward applications. The book originated as a collection of lecture notes and has benefited from fifteen years of classroom testing.

The author makes applications drive his exposition, but his goals are fairly broad. He wants students to learn to formulate a mathematical model, solve differential equations either analytically or numerically, analyze equations and solutions qualitatively, and interpret the results. Although the author promotes differential equations by emphasizing that they have applications in a great many fields, his emphasis through examples in the text is on uses in engineering and physics.

The text offers instructors a lot of flexibility in structuring their courses, with options for how much emphasis a topic will receive and what kinds of examples and applications are included. Although it was written to support a more advanced two-semester course, it could be adapted for a shorter and more basic first course.

The selection of topics is mostly traditional although there are occasional variations that introduce topics (the Bernoulli and Riccati equations, for example, as well as aspects of linear algebra like spectral decomposition) that don’t commonly appear in undergraduate texts. Numerical methods are introduced earlier than in other comparable books. The author often uses Maple, Matlab, Mathematica, or Maxima code in examples and believes it’s important that students become comfortable using one of these or an equivalent computer software package. He provides no introductory material for any of that software.

The discussion of Laplace transforms, while mostly standard, is more extensive than in comparable books and there are more examples, all worked out in some detail. The treatment of partial differential equations provides a very quick introduction and addresses mostly the heat, wave and Laplace equations.

Although the book is strongly oriented toward applications, the author does not ignore theoretical considerations and even proves theorems when he deems the proofs important for the student’s understanding. For example, he carefully states existence and uniqueness theorems and gives a proof using Picard’s successive approximation method.

Many of the pages of this book are dense with text, equations, tables, and plots, so much so that many pages look much too full. This is worse when there is computer code interspersed – sometimes cut in too close to the main text. Since this is already a very long book, changing the font or adding additional space would make it even longer. But it would definitely make it more readable and visually appealing.

This book is best suited for a yearlong course in differential equations for students in mathematics, physics, and engineering. It is particularly good with its many examples worked out in detail and its many exercises. The bibliography is quirky and has some interesting titles.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.


First-Order Equations. Applications of First Order ODE. Mathematical Modeling and Numerical Methods. Second-order Equations. Laplace Transforms. Series of Solutions. Applications of Higher Order Differential Equations. Linear Boundary Value Problems, Introduction to Green's Functions; Nonhomogeneous Boundary Value Problems; Singular Sturm--Liouville Problems, Bessel Series Expansion, Orthogonal Polynomial Expansions; Appendix: Software Packages. Answers to Problems. Bibliography. Index