
Many elementary texts on differential equations leave the student bewildered by the mass of special methods that are used for solving different types of equations. Also, accurate statements and rigorous proofs of existence theorems are usually avoided. As a result, the student acquires some skill and technique in handling the classical differential equations that occur in physics and elsewhere but he finds himself in a strange country when he takes a more advanced course that contains rigorous discussions.
The present text has overcome this difficulty by stressing in the first three chapters the geometrical and intuitive aspects, using lineal and circular elements, disks, and conical elements. Then the book in Chapters IV and V gives a rigorous treatment of existence theorems, using the method of successive approximations. No use is made of Cauchy's method of the calculus of limits and just a mention is included of the CauchyLipschitz method. The first three chapters lead up to the later chapters by their discussions of direction fields, of solutions in series, of the Wronskian and linear dependence. The method of successive approximations leads naturally into the Chapters VI and VII on interpolation and numerical integration and solutions. These chapters, by the way, are unusual in an elementary text. They are extremely well done, thanks to the author's own ability in and contributions to these subjects.
Attention should be called to the treatments of finite differences and of the symbolic operators, also to the emphasis placed on whole families of solutions. Chapter VIII on linear equations contains the Gramian, the Wronskian and linear dependence, as well as symbolic methods. The next chapter on certain classical equations gives a good introduction to the hypergeometric, the Legendre, and the Bessel differential equations. In Chapter X on partial differential equations of the first order the distinction between complete and general solutions is well brought out, also the geometrical interpretations of solutions are emphasized. This will help the student to grasp the subject better.
In the preface, the author states: "The partial differential equation of the second order is a vexatious problem for the writer of an elementary text. What has been attempted here has been the presentation of a compact, connected, and (it is believed) teachable body of material which exhibits those elementary methods of solution which are of commonest use." Here again the author has succeeded.
The book concludes with a good index. It is wellprinted, and its figures are interesting and very instructive. Many illustrative problems are worked out carefully in the text. Whereas many of the classical problems giving applications to geometry and physics are included, there are also to be found very modern and interesting problems such as the socalled parasite problem, the vibrating string and membrane, the conduction of heat, and the like. Some problems appear again and again to be solved by different methods or with different initial conditions. The early part of the book contains plenty of problems, well scattered throughout the chapters. The latter part of the book might have more problems than it has. Also a few answers now and then would help to give the student confidence.
There are only a very few misprints, such as ...
The style of the book is forceful, almost conversational, and decidedly refreshing. Many a teacher feels that if only an author could put into print his explanations to his classes, the result would be a good, teachable text. Unfortunately the language of writing is much more stilted than the language of speech. The author has come pretty close to writing as he would talk. Notice the use of the word except instead of unless at the bottom of page 115; also the sentence on page 173 that reads "A symbolic method in which the integrations are hitched abreast instead of tandem avoids this difficulty"; also such section headings as "Aids to Good Guessing" and "On the Making of Rules." Moreover it is well to note how the special cases are grouped under general methods with the comment that an energetic and intelligent student could keep on forever discovering new rules for integrating particular types of equations.
Parts of the text have been taught by the author to different classes of students. The whole text has been used by him in a year's course. A good semester's course could be offered using selected portions of the book and putting thereby the emphasis wherever the teacher should desire. Also the book could be studied without the help of a teacher. In every way this is a very good text on differential equations.
A D Campbell

Subsequent chapters cover special methods for equations of first order, linear equations of any order with a brief account of the use of the Laplace transform, solution in series of the hypergeometric, Legendre's and Bessel's equations, approximate numerical solutions, and two chapters on partial differential equations. Numerical solutions are preceded by a good chapter on finite differences, including approximate differentiation and integration and the algebra of operators î and E. General solutions of simple types of partial differential equations are obtained before separation of variables is used to solve problems of vibration and the Laplace equation in two dimensions. Linear dependence of functions receives considerable attention. It is perhaps a pity that the author does not illustrate the statement "that the vanishing of the Wronskian is not alone sufficient has been proved by an example for the case n = 2" on p. 87; at first sight this appears at variance with the theorem on p. 63.
The book may be recommended as a good blend of the scholarly and the practical approaches to its subject. The student reading without the help of a tutor will not find it easy. Although answers are given to all exercises, a larger proportion of worked examples in the text would be an advantage. ...
C G Paradine
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