Differential Equations as Models in Science and Engineering

This is a very unusual text in differential equations (both ordinary and partial) at the sophomore college level. The traditional approach to the subject defines differential equations, gives examples, discusses solution methods and then points out applications to other areas of science. In this book, by contrast, the science comes first and is used to motivate the differential equations: a scientific problem is posed and analysis of it leads to a differential equation. Once this differential equation is in hand, the underlying science is used to analyze the equation and to interpret its solutions.

In short, what we have here is a very applied introduction to the subject: all the topics in differential equations that are covered in this text are seen through the lens of applications. Even the table of contents reflects this shift in emphasis: instead of seeing sections titled “Variation of Parameters” or “Exact Equations”, as we would expect to find in a differential equations text, we find sections here titled “A simple chain of chemical reactions” or “LCR circuit with constant applied voltage”.

Chapter 1 discusses linear ODEs of first and second order. The chapter begins with problems (e.g., population growth and decay, simple electric circuits) that lead to simple first order linear ODEs with exponential solutions. It is then shown that modifying the assumptions of the problem leads to the introduction of forcing terms in the differential equation. After this, problems calling for second order ODEs (such as compound chemical reactions) are discussed.

Chapter 2 looks at more complicated forcing terms, specifically periodic ones. These lead to a discussion of Fourier series. After this, Chapter 3 considers two-point boundary value problems for linear second order ordinary differential equations. These are motivated by models of quantities that vary both spatially and in time; however, for this chapter, an assumption is made that the quantity has reached steady state and can be assumed to be not time-dependent.

This assumption is removed in the next chapter, where quantities are now assumed to vary as a function of both time and position. The presence of two variables, of course, leads to partial differential equations. The “big three” examples of second order linear PDEs (diffusion equation, wave equation, Laplace equation) are introduced — again, via specific problems that lead to them — and studied, primarily using separation of variables as a solution technique. (The content of the previous two chapters plays a significant role in this method.) Interestingly, the motivating example for the wave equation is the propagation of signals along a cable; the traditional example of the motion of a vibrating string is not discussed.

The final chapter of the book discusses systems of ODEs. The first section of this chapter isn’t really about systems at all, but is about single nonlinear first order ODEs; it is here that we finally get a discussion of topics like variables-separable equations that are often taught much earlier in a differential equations course. When discussing systems, the author mostly limits himself to linear equations; only about ten pages is spent on systems of nonlinear first order equations, with the predator-prey situation given as the primary model.

As this summary of the contents makes clear, there are some topics that are often discussed in a first-semester ODE course that are not covered here. Laplace Transforms, for example, are conspicuous by their absence. Neither is there any discussion of numerical methods for differential equations. Students also find interesting stories about how bridges have collapsed because of issues involving resonance that can be explained with ODEs, but this is not mentioned here either. Systems of differential equations are not introduced until the end of the book (thereby disappointing instructors who, like me, think the dynamical systems point of view is valuable and should be introduced as early as possible) and are not covered to the extent that many professors might like. Very little attention, for example, is paid to nonlinear systems or chaos.

The mathematical prerequisites for this book have been kept to a minimum; a background in linear algebra, for example, is not assumed. The author develops what linear algebra he needs on an ad hoc basis, typically limiting himself to the \(2\times 2\) case. What is necessary, however, is a pretty decent background in engineering or physics.

The author’s writing style is competent but fairly dry, and often struck me as unnecessarily wordy. Consider, for example, his discussion of the separation of variables method for PDEs. In this method, one assumes that a solution \(u(x,t)\) can be written as a product of a function of \(x\) and a function of \(t\); this then leads to an equation of the form \(X(x)=T(t)\); i.e., a function of \(x\) alone is equal to a function of \(t\) alone. At this point, it is generally observed that the only way this can happen is for both functions to be constant. Most PDE books spend a sentence or two on this trivial observation: pick a certain value of \(x\), plug it into the left hand side, and observe that the right hand side is then a constant for all \(t\). Baker spends about three paragraphs discussing this issue, and even then, he buries, in a footnote, the simple explanation just given in the last sentence. (An even simpler observation, which the author does not make: differentiate the left hand side with respect to \(x\) and note that the derivative must be \(0\), so the function is a constant.)

As another example of a wholly unnecessary statement, the author, at one point, has occasion to consider the equation \(y'(t)=0\) and points out that this has a constant function as a solution. Then the author feels the need to add the observation “Of course, it is questionable whether [this equation] really is indeed a differential equation rather than just a specification of a derivative with an obvious anti-derivative.” I see nothing to be gained by a comment like this, other than confusion. Is the author really trying to suggest that a simple equation like this one is not really an example of a differential equation? Is the equation \(y'(t)=t\) not really a differential equation, because the anti-derivative of the function \(t\) is also quite obvious?

While some parts of the book are unnecessarily wordy, other parts are not wordy enough. For example, several standard techniques that provide an easy introduction to solving differential equations are omitted. After arriving at the simple differential equation \(y'(t)=\lambda y(t)\), for example, the author does not mention that this is a separable equation that can be solved by the usual trick; he instead finds an exponential solution based on physical considerations. In other words, he guesses an exponential solution and then verifies that it works. Had the author just integrated both sides of an equation, he could have shown that a solution to this equation had to be an exponential and therefore established the uniqueness of a solution taking on a given initial value. Instead, the author shows uniqueness by the much more complicated method of showing that the terms of the Taylor series of a solution are uniquely determined. (The idea of a variables-separable first order equation is not mentioned until the last chapter of the book, on linear systems.)

Another omission in this text is the complete lack of any bibliography or list of further references. I view this as a serious problem in any undergraduate mathematics textbook. Also, the Index seemed weak: a person looking for a discussion of “separation of variables”, for example, will not find it under “S” or “V”, but can find it under “Partial Differential Equations” or “Boundary Value Problems”; I found a reference to “variation of parameters” not under “V” or “P” but by first going to “Linear, First-Order Differential Equations”, then looking at the sub-entry “particular solutions”, then looking under that for “variation of parameters”.

I also frequently found myself wishing for things to be more precisely stated. As an example, the author lists 29 “principles” throughout the text, each of them a word (or paragraph) of wisdom about differential equations. There is nothing wrong with this (in fact, it could be a very good idea), but some of these principles are worded so imprecisely as to be confusing. Principle 2 states, for example, “The solution to a differential equation is a function that should have the appropriate properties so that any term in the equation makes sense.” I would have a lot of sympathy for a student who has difficulty figuring out just what this is supposed to mean — whether a term in the equation makes sense ought not depend on any particular solution. Based on context, I think the author is trying to convey the idea that a solution should have as many derivatives as necessary to satisfy the equation, but the principle itself is too vaguely worded to allow this interpretation on its face.

As an another example, the author states on page 120 that the “only basic functions that exhibit periodicity are the sines and cosines”; it is not until a few pages later, on page 126, that he acknowledges that constant functions are also periodic, but calls this an “artificial case”. He also defines the period of a function to be the “smallest possible period” but fails to mention that there is even an issue as to whether there must be a smallest possible such number.

Likewise, definitions are not stated with the degree of precision that I would expect to see in a textbook that will be read by mathematics majors. I’m not even sure that I ever saw a specific definition actually set off and denominated as such. (Likewise, the only theorem that I can recall being labeled as one is the Fourier Convergence theorem.) On page 8 it is mentioned that a certain ODE is linear, and the author explains that that means “the unknown function and its derivatives appear separately and not as the argument of some other function”. This seems unclear to me, and actually wrong: the ODE \(y+y'=0\) is certainly linear, but the unknown function and its derivative “appear as the argument” of the function \(F(a,b)=a+b\).

I suspect that the author’s looseness with language derives from the fact that he is addressing engineering students rather than mathematics majors. It is noteworthy that he has dedicated this book to his engineering students, who, he says, “strongly influenced” the book and who “are attracted to the power of mathematics rather than its beauty.” It seems significant here that the author says “rather than”, not “in addition to”. This reflects an attitude of ignoring anything that is not related to the applications of differential equations. This is an attitude that I certainly would not want to indulge in any differential equations course that I taught, particularly not at the expense of the many math majors who pass through such a course and who will not wind up becoming scientists, engineers or applied mathematicians. Why can’t a good introductory ODE course teach both the power and the beauty of the subject? (Other books do, including Simmons’ Differential Equations with Applications and Historical Notes, and Differential Equations by Blanchard, Devaney and Hall.)

The author’s “applications first, last and always” point of view, I think, creates problems in other ways as well.

First, the book is weak on the kind of routine drill examples that, I think, the current generation of college students really needs in order to be successful in such a course. The examples that arise are those that come up in physical modeling problems, and there are comparatively few of these. Moreover, they tend to be somewhat more algebraically complicated than the simple examples one finds in introductory ODE books. Even in the first chapter, for example, at a time when a student needs simple examples, he or she is forced to plow through algebraic calculations that even the author acknowledges to be tedious.

Second, I think that the author’s approach increases the level of difficulty of the learning process. My preference is to get the students comfortable with the material before plunging into applications of it; one has to learn to crawl before one can walk. I think that starting off the semester with a simple applied example, just to see how differential equations arise in real life, is a splendid idea, but once that has been done it seems advisable to me to work with the ODEs for awhile and get familiar with them before investigating further examples. If nothing else, students with weak science backgrounds, who may not be comfortable with the physical discussions underlying the development of the differential equations, can avoid having yet one additional hurdle placed on their road to understanding. For this reason, I think many students may find this book to be less reader-friendly than, say, Simmons or Blanchard, Devaney and Hall.

To summarize and conclude: an instructor teaching a course to engineering students only who shares the author’s view that applications trump everything else, may find this book worth a look. Those who teach introductory ODE courses that are also attended by mathematics majors, and who see the need to do things other than applications, should likely look elsewhere.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.