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The following review appeared in the

Handbook of differential equations

by Daniel Zwillinger

Academic Press, Inc., Boston, MA, 1989, xx+ 673 pp., $49.95.

ISBN 0-12-784390-6

Reviewer: Ray Redheffer (University of California, Los Angeles)Reviewers for this journal are invited to give a somewhat discursive account, and my review will not deviate from the custom so established. I taught undergraduate differential equations for the first time at Harvard University in 1948 and have taught it more or less regularly ever since. Added to that are about a dozen graduate courses on differential inequalities here and abroad. A few thoughts arising from this experience are summarized first. Next comes a survey of the current scene as reflected by popular texts, and finally Zwillinger's book.

For those who, like me, prefer to start a detective story by peeking at the end, I can save time by saying that Zwillinger's book is an outstanding scholarly achievement, original in both plan and execution. It will surely appeal to all who have more than a passing interest in differential equations. If you have read this far, you probably qualify; so you might as well skip the review and get the book.

General remarksOne of the charms of elementary differential equations is that most of the proofs are easy, yet they yield results of obvious importance. For example, to establish uniqueness for the solutions of

y''+p(t)y'+q(t)y=f(t),y(t,_{0})=y_{0}y'(ton an interval containing_{0})=y_{1}t, let_{0}w=uwhere^{2}+(u')^{2}uis the difference of two solutions. If the coefficientspandqare bounded,wsatisfies a differential inequality that impliesw=0and uniqueness follows. A similar method gives uniqueness for both linear and nonlinear equations of arbitrary order.The importance of uniqueness can hardly be overestimated. If a differential equation arises from a physical system, it is only by uniqueness that we can say that our solution represents

the behavior of the system, not merelya possible behavior of the system. On the theoretical side, uniqueness for thenth-order linear equation underlies the entire theory of the Wronskian and linear independence.Here is an example pertaining to existence. If a function \phi has a bounded derivative on a finite interval

(a,b), it is easy to show that the endpoint limits \phi(a+)and \phi(b-)both exist. This allows us to extend a local to a global solution without the familiar complications entailed by an oscillatory approach to the boundary. Again the proof is easy, the ramifications significant.Also related to existence is the fact that convergence of power-series solutions can be established by an easy mathematical induction; the method of majorants is not needed. For boundary-value problems we now have Chernoff's novel approach to convergence of Fourier series. When extended to piecewise smooth functions, it opens up a subject that until then was almost forbidden territory.

If the theory of exact differentials \omega

= P dx+Q dyis developed by use of definite integrals, as it should be, one can readily see why the main result might be expected to fail for ring-shaped regions. Thus the way is prepared for simple connectivity and line integrals as encountered in more advanced courses.The familiar theorem asserting linear independence of

{tis both important and easy to prove. It yields a basis for solutions of the general linear equation with constant coefficients, it plays an essential role in the theory of asymptotic stability, and it is a logical prerequisite for the method of annihilators. From a broader perspective,^{j}e^{st}}elementary differential equationsis perhaps the first place where students learn to use complex numbers with skill and confidence.These remarks may serve to emphasize the fertile interaction between easy proofs, uncomplicated ideas, and significant results, that is offered by the study of differential equations. The interaction provides an ideal introduction to more austere courses of modern algebra and advanced analysis, and it has the potential to make a significant contribution to mathematics education.

On the world scene, this potential is actually realized. For example Wolfgang Walter's marvelously brief and elegant text proves virtually everything, even the spectral theorem for compact self-adjoint operators! Yet it has been, for a long time, the most popular undergraduate text on differential equations in Germany.

Current trends as set by textbooksReturning to reality, let us look at some of the most popular texts currently used in this country. Typically, key formulas and theorems are highlighted by colored print, or displayed in boxes, so students can do the homework without reading the book. The text itself is characterized by the omission of proofs ``without apology,'' as is sometimes said, or by the boast ``no frills'' which really means ``no proofs.''

For example, what about uniqueness? One strategy is to discuss the second-order linear equation with positive constant coefficients, by a proof that does not generalize, and let it go at that. Another strategy is to discuss Picard iteration in the one-dimensional case, typically through a mixture of text and problems in which uniform convergence is barely mentioned. It is then said, without further details, that the alleged ``proof'' extends to vectors and thence to higher-order equations. But closer examination indicates that uniqueness has not really been proved even for the second-order linear equation. Therefore the consequences of uniqueness have not been proved either.

Global existence and convergence of series solutions are omitted from nearly all popular books. However, one much-used text deals with the convergence of Fourier series in the following enterprising fashion: By partial integration, it is shown that the coefficients of a

Cperiodic function are of order^{2}1/n, hence the series is uniformly convergent. This completes the proof. The question whether the series converges to the function that generated it is not addressed at all.^{2}More surprising is the fact that the theory of exact differentials is commonly based on indefinite integrals, the equation

\frac{\partial}{\partial y} \int f(x,y) dx=\int f_{y}(x,y) dxbeing used as a disguised form of

F. The trouble with this approach is that it is local. If students are asked why the proof fails for a ring-shaped region, they cannot answer, because the role of simple connectivity is effectively concealed. Yet a development by definite integrals can be found in books written thirty or more years ago. Retrogression is not progress._{yx}=F_{xy}In most popular books linear independence of

{tis established only for two functions, or for the case^{j}e^{st}}j=0, or sometimes in an additional case in which all exponentssare equal. But the applications mentioned above require the general theorem; special cases are not enough. If the theorem is unproved, its consequences are also unproved.Confident and skillful use of complex numbers is seldom seen in popular texts, with the result that our math students are kept far behind those in physics and engineering. In fact at least one well-known book casts doubt on the familiar formula for differentiating exp

(a+ib)tand suggests that results obtained with complex numbers must be checked by use of the corresponding real form! Yet justification of the key formula is trivial; merely differentiate the defining relations

ecos^{ibt}=bt+isinbt,e^{(a+ib)t}=e^{at}e^{ibt}.To paraphrase Madame Roland: O rigor! what things are done in thy name!

Since complex numbers are avoided as far as possible, students typically are asked to substitute an expression like

tcos^{2}e^{at}(Abt+Bsinbt)into a high-order differential equation to get the constants

AandB. Even more typically, they are asked to write the form of aa trial solution and let it go at that. Anyone faced with a genuine problem in mechanical or electrical engineering may be forgiven for doubting the utility of this long-outmoded procedure.

Zwillinger's bookWith these remarks as background, let us look at the book under review. Since proofs are systematically omitted, it might be thought that this reviewer would object. But, not so. Zwillinger's aims are entirely different from those set forth above. His book is more like Pierce's

Table of integralsthan like a conventional text. The emphasis is quite properly not on proofs, but on correct statements, and there is no attempt to give an appearance of rigor without the reality.Within this self-imposed limitation, what a wealth of interesting results in small compass! Zwillinger makes consistent use of the format:

The first item describes the kind of problems the method pertains to. If your problem fails to qualify, read no further. The second says what the method will do. The basic idea is explained next, and finally comes a description of the procedural details. A most happy feature of the exposition is that every method is illustrated by at least one well-chosen example. For example, bifurcation is illustrated by a bead on a ring that rotates about a vertical diameter. Equating forces makes it clear why there will be two equilibrium positions when the rotation is rapid, only one when it is slow. The differential equation then allows us to decide about stability. Equally impressive is an illustration of interval analysis in connection with Picard's method. One example like these is worth many words.

- applicable to
- yield
- idea
- definitions and concepts
- procedure
The book must have involved an immense amount of labor, and as a consequence, it has far too much content to be summarized here. For example, section 33 entitled ``Look Up Technique'' contains about 250 equations with names and references, and the bibliography for this section alone has 112 references. Here is a list of the main headings in the Table of Contents. On the average, each is accompanied by about thirty subheads:

- Definitions and concepts
- Transformations
- Exact analytical methods, ODEs
- Exact analytical methods, PDEs
- Approximate analytical methods
- Numerical methods: concepts
- Numerical methods for ODEs
- Numerical methods for PDEs
An important feature is that the book is typeset by the author himself, using

TeX. No doubt this helps to make the relatively low price possible, and it eliminates the usual error-generating transfer from manuscript to printed page. (I found only two misprints, both unimportant.) The attractiveness of the typography suggests that the author is a skilled compositor as well as a skilled expositor.It is to be expected, of course, that every reader can come up with favorite topics that are excluded. I should have liked to see LaSalle's improvement of Liapunov's method, a better development of the matrix Riccati equation and its bearing on scattering and transfer, a more up-to-date account of maximum principles, the concept of the graph of a matrix and its relation to Lotka-Volterra systems, and further details about monotone operators, to mention five examples. Also virtually no methods are discussed in depth. But if more topics had been included, and each had been fully developed, the book would have been over 7000 pages long, not less than 700. The author's decision to omit proofs is dictated by this same requirement of reasonable size.

One aspect of the exposition to which I take exception is Zwillinger's preference for expository accounts and (in my view undistinguished) elementary texts over original sources. Too often the treatment in these references is decidedly superficial, and they also distort the historical record. For example, the main sources for the line method in parabolic differential equations are all omitted.

I think the book could be a good supplement to a course on differential equations, but, as is clear from the early part of this review, I would be aghast if it were the main text. That would be like using a book of cooking recipes as the main text for organic chemistry. If we mathematicians abandon the goal of logical development, who will replace us?

The following review appeared in

Zwillinger, DanielHandbook of differential equations

1989

Academic Press, Inc., Boston, MA, 1989, xx+673 pp, $49.95.

EnglishFrom the introduction: ``This book is a compilation of the most important and widely applicable methods for solving and approximating differential equations. As a reference book, it provides convenient access to these methods and contains examples showing their use.

``The book is divided into four parts. The first part is a collection of transformations and general ideas about differential equations and how they are solved. This section of the book describes the techniques needed to determine whether a partial differential equation is well posed, what the `natural' boundary conditions are, and many other things. At the beginning of this section is a list of definitions for many of the important words describing differential equations and their solutions.

``The second part of the book is a collection of exact analytical solution techniques for differential equations. The third part of this book deals with approximate analytical solution techniques. For the methods in this part of the book, the format is similar to that used for the exact solution techniques.

``When an exact or an approximate solution technique cannot be found, it may be necessary to find the solution numerically. Other times, a numerical solution may convey more information than an exact or approximate analytic solution. The fourth part of this book is concerned with the most important methods for finding numerical solutions of common types of differential equations. When possible, a short FORTRAN program has been given. At the beginning of the fourth section is a brief introduction to the terms used in numerical methods. In that section, terms are defined and some general references are given.''

The following review is from the

Zwillinger, Daniel, 1989.Handbook of differential equationsNew York: Academic. xx+ 673 pp, $49.95. ISBN 0-12-784390-6If the term ``handbook'' suggests to you seemingly endless lists of equations, solutions of equations, relations among solutions, identities concerning solution, and so forth, do not be put off; this is not that kind of handbook. Rather, this is a handbook of methods for solving ordinary or partial differential equations.

There are 180 numbered sections, of which about 40 deal with definitions and introductory or explanatory material. The remaining ones are devoted to methods, usually one per section. For each method, the author gives the type of equation to which it applies, the anticipated output, and its basic idea. Then there is a brief explanation of how the method works, illustrated by one or more examples, and concluding with a short list of references. For newer methods, the references often include the original papers; for older methods, the references are to recent textbooks or monographs.

The material is organized into four chapters; Definitions, Concepts, and Transformations (114 pp.), Exact Analytical Methods (270 pp.), Approximate Analytical Methods (121 pp.), and Numerical Methods (142 pp.).

The information content of the book is very high., and its organization makes it easy to locate material on a particular topic. The emphasis is on practical methods that have been and are being used, especially ones that have proved to have interesting applications.

Of course, no single book can contain everything one might want to know about differential equations; and the contents of this one reflect the experience and interests of the author. As the chapter headings indicate, the book has much to say about classical and modern analytical methods, and quite a bit about numerical techniques, but much less about geometrical or topological approaches. Except for the appearance of the Lorenz equations in a list of named equations, there is nothing on strange attractors, chaotic solutions, and related mattes of current interest.

Although some of the methods are indeed elementary, others require considerable mathematical sophistication. Most ofthe book should be accessible to seniotrs or graduate students in various disciplines, or to practicig engineers or scientists. Anyone with prior experience with differential equations who wants a compendium of current methods will find this book useful. Instructors might use it also to supplement the text in a cource, or as a source of topics for independent study or student research projects. Even experts may benefit from it, if only as a refresher or as a place to browse for ideas.

The author has produced a book that is significantly different from anything else on the market. Its wide-ranging coverage, intermediate level, and straightforward style should make it attractive to a correspondingly wide range of readers.