I teach differential equations regularly and I have yet to find a textbook that can be used for both the Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) courses. Combining ODEs and PDEs in a single accessible volume is an extremely difficult task, as evidenced by the lack of such texts. In the book under review, Henner, Belozerova, and Khenner cover most of the fundamental topics found in introductory ODEs and PDEs courses, nicely balancing scope without sacrificing content.
The book has fourteen chapters and is separated into two parts. Part I covers ODEs, Boundary Value Problems, Fourier Series, and an introduction to Integral Equations; Part II covers second order PDEs. These two parts divide the book roughly in half (by page count) even though there are nine chapters in Part I and five chapters in Part II. Included with the text is a software tool (cd) designed specifically to be used with it. In addition, five appendices serve as references for various topics, including one on how to use the software.
Chapters 1, 2, 3, 5, and 8 comprise the core topics typically found in an introductory undergraduate ODEs course. Chapters 4, 7, and 9 provide the main ideas for transitioning from ODEs to PDEs and could be included along with chapters 10–14 to form the main topics for a PDEs or graduate level differential equations course. The authors have managed to provide the right amount of details and have outlined the text in such a way that all material needed to solve the PDEs discussed in Part II can be referenced within the text. This, in my opinion, is the main strength of the book.
My impression is that the book was originally about PDEs, with the ODEs added as an afterthought. Part II is rich in development, examples, applications, and includes many graphs to help the reader with the material. Part I does include the main topics typically found in an ODEs course, but lacks motivation, examples, and explanations. So this is a good book for introducing second-order PDEs, but I would be reluctant to recommend it as an introductory ODEs textbook by itself. I would be willing to use this text to teach an upper level differential equations course, as referencing introductory ODEs material is straightforward and self-contained. That being said, this single book could be used successfully for a series of differential equations courses that covered both ODEs and PDEs if the same students took the courses.
Part I of the text is technique-oriented, giving an equation (usually motivated by a single sentence suggesting that the equation arises in applications of physics, engineering, etc.) and applying a specific technique to solve. Actual references to application problems are few and far between. For example, the first real mention of applications comes in section 1.8 after first-order methods have been introduced (separation, exact solutions, homogeneous substitutions, variation of parameters, etc). Unfortunately, the three examples given in section 1.8 are all separable equations and do not help motivate the reason for studying the other methods. A similar lack of motivating examples is common for the remaining topics in Part I.
In my experience teaching introductory ODEs at three different universities, students taking ODEs often need motivation for studying solutions to ODEs. If an instructor has enough experience to provide motivating application problems and to fill-in needed details, then this book could be used for an ODEs course. In that case, I would recommend that the chapters on Laplace Transforms (6) and Series Solutions (8) be further developed. In their current form, these two chapters lack basic examples and development that would be valuable to students, especially those studying engineering.
In Part II the wave, heat and diffusion, Laplace, and Poisson equations are all developed in an accessible and clear manner. Emphasis is placed on exploring each of these equations under various initial and boundary conditions, and the authors do a good job of providing a comprehensive collection of examples. In most exercises throughout Part II, analytical solutions are encouraged and the software program is referred to when integrals are not easy to evaluate. It is clear that the authors spent a lot of effort in making Part II of the text interesting and relevant. In many solution methods, references to ODEs solutions methods, Sturm-Liouville problems, and Fourier series help connect the general theories of ODEs and PDEs.
For a first edition the book seems relatively free of major typos and editorial problems. The main issues I encountered were inconsistently used fonts (distinguishing between examples and general discussion), difficult to follow sentence structure and grammar, and in some cases, the use of ideas before they were properly defined (e.g., self-adjoint operators, delta function). The authors also used language that seemed out of the norm such as “plugging the general solution in the initial conditions results in…” (p. 45). Although I found these issues mostly in the first part of the text, overall they did not detract too much from the main ideas. The software included with the text did not work on my Mac but worked well on a windows machine.
In summary, the task of providing a comprehensive text on ODEs and PDEs is a difficult one, simply because of the amount of content that can be covered. This text finds a nice balance between general topics of ODEs and second order PDEs. The authors’ choice of omitting nonlinear systems of ODEs and first order PDEs is not unreasonable; it does not diminish the value of the book unless the reader or instructor prefers to cover these topics. The lack of applications and motivation in Part I of the text does make it a difficult choice as a stand-alone book for introductory ODEs. However, it could be used for a single PDEs course or in a sequence of differential equations courses where a survey of second order PDEs is desired.
Joe Latulippe (firstname.lastname@example.org) is an assistant professor of mathematics at Norwich University. He is a Project NExT Fellow (Sun Dot ’07), and is interested in mathematical biology and modeling. He is also an assistant coach for the Norwich University Men’s Lacrosse team. In his free time, Joe trains in Aikido (a non-violent Japanese martial art), and enjoys painting and drawing.