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Ordinary Differential Equations

William A. Adkins and Mark G. Davidson
Publication Date: 
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Undergraduate Texts in Mathematics
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

Here at Iowa State University, the traditional sophomore-level “diffyQ” course comes in two versions: a three-credit course (Math 266) that does not cover Laplace Transforms (hereinafter, LTs), and a four-credit version (Math 267) that does. But for this one topic, there is no structural difference in the courses, and indeed some sections of them meet simultaneously for a part of the semester, with 266 ending before 267. The textbook for both courses is, therefore, the same, and, like most differential equations texts, it covers LTs near the end, so that this topic can be omitted if desired.

Not this book, however. The major distinction between this book and the existing textbook literature is that it introduces LTs quite early (in chapter 2, in fact), and actually uses them throughout the book. For example, they play a big role in the authors’ treatment of linear differential equations of degree n with constant coefficients. Such an equation can be written as q(D)(y) = f where D is the differentiation operator and q is a polynomial of degree n; as is well known, if f is the zero function, the general solution to this equation is a vector space. The authors use LTs to define the standard basis (which they call ℬq) for this space. A bit later in the book the authors use LTs to transform some second order linear ODEs with non-constant coefficients into first order ODEs. Still later in the text, connections between LTs and matrices are established; the matrix exponential eAt, for example, is the inverse LT of the resolvent matrix A.

On those few occasions, many years ago as a graduate student, when I taught ordinary differential equations, I discussed LTs only briefly and only as one more in a list of mechanical solution techniques: starting with a differential equation, such as, say, y'' + 4y = 4x, with y(0) and y'(0) specified, I would apply the LT ℒ to both sides of the equation and, using previously stated facts about the relationship of ℒ{y''} and ℒ{y'} to ℒ{y}, would convert the differential equation into an algebraic equation in the variable ℒ{y}. After solving this algebraic equation, use of a table of inverse LTs would allow us to solve for y. The whole process struck me not only as mechanical but, to be blunt about it, fairly tedious and not terribly interesting; it wouldn’t surprise me if I conveyed to the class a distinct lack of enthusiasm at this point of the course. What I didn’t realize at the time was that Laplace transforms could (as described above) also be used to shed some light on other aspects of the course that were covered long before we got to this topic. A book like this one would have prevented some of these misunderstandings. Even if I didn’t present the material that way in class (and doing so, I think, would have somewhat increased the level of sophistication of the course), it would have been nice for me to have known these facts.

Of course, there are going to be a number of teachers of sophomore ODE courses who, either because of personal preference or (as at ISU) the demands of a syllabus, do not want to introduce LTs early. The natural question then becomes whether this book can be used in a way that suits their needs. The authors’ own chapter dependence chart suggests a negative answer, because it lists chapter 2 (the one introducing LTs) as necessary background for most of the subsequent chapters (the one exception being chapter 8 on basic matrix algebra). This dependence chart notwithstanding, I had the impression that an instructor could, with some effort, navigate a course through the book that avoided LTs. However, it does not seem to me that the benefits of doing so would outweigh the effort. Books on this subject are not exactly scarce and, with the exception of the unusual feature of introducing LTs early, this book follows a fairly conventional path through the typical basic sophomore ODE course: basic solutions methods for categories of first order ODEs, second order and then nth order linear ODEs with constant coefficients, second order linear ODEs with non-constant coefficients using such methods as variation of parameters and the Wronskian, power series methods, and systems of linear ODEs (including matrix exponentials). The writing is competent and clear, but not particularly memorable; there is good attention paid to applications of the subject but, again, this is not an unusual feature among texts of this nature. So, there is no compelling reason to choose this text unless you are prepared to buy in to the authors’ philosophy regarding LTs. If you do buy into that philosophy, or are willing to be persuaded to do so, this book is tailor-made for you.

The exercises in a mathematics book of this level are always important, and this book does a good job with them. There are quite a lot of them, mostly of a fairly computational nature; a full solutions manual of almost 350 pages is available to adopters (and at least one reviewer), and there is also a student solutions manual for the odd-numbered problems that is, not surprisingly, just about half that size. There are also a number of solutions to odd-numbered problems in the back of the book (about fifty pages worth).

I do have a couple of quibbles, one of them not the responsibility of the authors and the other something that may be more of an issue to me than to others. Taking the latter first, in the chapter on separation of variables, the authors tell the students to “multiply by dt”, which to me (at least at this level) is like hearing squeaky chalk on a blackboard. I do realize that the theory of differential forms actually does allow the expression dy/dx to be treated as a fraction, but this comes far later in a student’s mathematical career and many calculus textbooks make a point of saying that the expression is not a fraction. Different books treat this issue in different ways. The popular ODE textbook by Blanchard, Devaney and Hall, for example, manages to address the subject in a reasonably precise and informative manner; they acknowledge that the reader “probably became nervous at one point” by treating dt as a variable and then proceed to explain “what is really going on” in terms of the chain rule. But perhaps I am nitpicking here; I suspect many of my colleagues wouldn’t object at all to just “multiplying” by dt in a course at this level.

As for the first of the two quibbles that I mentioned, there are a large number of blank pages in this book. I didn’t make any attempt to count them all, but from a quick estimate I would say the number of them was in the vicinity of 75 or more. There are also a great many other pages that, although not completely blank, contain very little writing. One would think that the publishers of an 800-page book would be making an effort to keep the size down and thus make it less cumbersome, but instead about ten percent of a book that is already quite thick and unwieldy is simply blank. I found this annoying and distracting.

But these, as I said, are quibbles. My overall impression of this book is quite positive: I enjoy books with a novel and interesting point of view, and, even though this is an elementary subject, I learned things from this text that I did not know before.

Mark Hunacek ( teaches mathematics at Iowa State University.

Preface.- 1 First Order Differential Equations.- 2 The Laplace Transform.- 3 Second Order Constant Coefficient Linear Differential Equations.- 4 Linear Constant Coefficient Differential Equations.- 5 Second Order Linear Differential Equations.- 6 Discontinuous Functions and the Laplace Transform.- 7 Power Series Methods.- 8 Matrices .- 9 Linear Systems of Differential Equations.- A Appendix.- B Selected Answers.- C Tables.- Symbol Index.- Index.