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An Introduction to Ordinary Differential Equations
"Written in an admirably cleancut and economical style." — Mathematical Review. A thorough, systematic first course in elementary differential equations for undergraduates in mathematics and science, requiring only basic calculus for a background, and including many exercises designed to develop students' technique in solving equations. With problems and answers. Index.
320 pages, Paperback
First published January 1, 1961
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May 29, 2023
This reviewer must confess to an oversight in his undergraduate education: he satisfied himself with their treatment in physicists’ textbooks and never took a proper course on ordinary differential equations. Where to turn, now that at a more mature stage in his career he wishes to remedy the defect? After all, there are a multitude of textbooks in the subject at the beginner’s level. Upon consideration, he settled on Earl A. Coddington’s An Introduction to Ordinary Differential Equations (1961), now available in a convenient Dover reprint. Why? First off, one has a right to expect clarity from a pure mathematician. At 292 pages including answers to exercises and index, moreover, the present work has the virtue of being succinct. Unlike another popular text by Morris Tenenbaum and Harry Pollard that runs to over eight hundred pages, Coddington does not aim at comprehensive coverage of applications yet manages to go just about as far in treating the elementary theory.
Right off the bat, one appreciates Coddington’s lucid and crisp style. After preliminaries in chapter zero one might well skip and demonstration of some simple ideas at the foundation of the subject in chapter one, chapters two and three get down to a nice and systematic development of differential equations, proceeding from the case of constant coefficients to variable coefficients and from homogeneous to inhomogeneous (theory of Wronskians and the method of finding a particular solution). Everything is proved satisfactorily, for instance the annihilator method of section 2.11 exhaustively justified in section 2.12. The case of an n-th order linear equation follows in section 2.7 directly from the preceding material.
Chapter four takes up the subject of regular singular points (when the coefficient of the highest derivative passes through zero). This offers a chance to illustrate the method of solution by power series, since many basic equations of mathematical physics assume the form of a second-order differential equation with regular singular points. The homework exercises take one through several of them (Bessel equation, Legendre equation, Laguerre equation, hypergeometric equation). Then chapter five circles back and offers an efficient proof of the existence of a solution to first order differential equations under a Lipschitz condition (as Coddington explains, some condition beyond continuity is needed). It comes down to formulating an integral equation that the solution has to obey and from it constructing a sequence of successive approximations, the convergence of which follows readily from the Lipschitz condition. Chapter six concludes by covering the existence and uniqueness of solutions to systems and n-th order equations.
About the approximately three hundred homework exercises strewn liberally throughout the text: many problems turn out not to be especially hard, they are often arranged in sequence so that having solved the earlier ones one knows what to do for the later. To proceed with dispatch, use Mathematica to perform the integrations or find roots of indicial polynomials or carry out substitutions for you (since this aspect of the derivation is not of the essence). For it comes down to one’s pedagogical philosophy: if one be confident in his grasp of the calculus little purpose would be served by seeking to work out the solutions without the aid of software. The hints can be almost too complete and thereby deprive one of a sense of accomplishment in figuring out on one’s own what to do (they basically sketch the solution for you). Nevertheless, though the problems are often straightforward they do illustrate concepts it is important to know (for instance, a problem on partial fraction decompositions). Thus, one should attempt all the problems. If routine, they help to internalize what one has just read; if perplexing (about one in every ten), they force one to think harder for an hour or two and thereby afford an ideal learning opportunity. But none remotely approaches the legendary difficulty of many of the problems in baby Rudin (for our review of this classic textbook in introductory analysis, see here).
The treatment in the present work leans to the theoretical, with the consequence that the subject begins to seem orderly, more so than anticipated in view of its reputation as but a grab bag of techniques lacking full generality (as say Courant and John portray it in the second volume of their course on calculus, our review here). Though often billed as being on the advanced undergraduate level, the present text is not perhaps quite so difficult. Everything in the exposition follows logically without too much effort, all of the problems that could pose a challenge are supplied with generous hints anyway and only occasionally do they call on tools from introductory analysis (to show convergence of a series) – raising the danger that the careless student might be lulled into a sense of complacency: for graduate-level textbooks certainly will expect the student to bring more of his own wit to the table and to display initiative.
Recommended for the serious student of pure mathematics looking for a competent guide into the subject of ordinary differential equations, without wanting to be detained by an overlong tour. For a more advanced, definitely graduate-level treatment see the standard reference by Coddington himself, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).
Right off the bat, one appreciates Coddington’s lucid and crisp style. After preliminaries in chapter zero one might well skip and demonstration of some simple ideas at the foundation of the subject in chapter one, chapters two and three get down to a nice and systematic development of differential equations, proceeding from the case of constant coefficients to variable coefficients and from homogeneous to inhomogeneous (theory of Wronskians and the method of finding a particular solution). Everything is proved satisfactorily, for instance the annihilator method of section 2.11 exhaustively justified in section 2.12. The case of an n-th order linear equation follows in section 2.7 directly from the preceding material.
Chapter four takes up the subject of regular singular points (when the coefficient of the highest derivative passes through zero). This offers a chance to illustrate the method of solution by power series, since many basic equations of mathematical physics assume the form of a second-order differential equation with regular singular points. The homework exercises take one through several of them (Bessel equation, Legendre equation, Laguerre equation, hypergeometric equation). Then chapter five circles back and offers an efficient proof of the existence of a solution to first order differential equations under a Lipschitz condition (as Coddington explains, some condition beyond continuity is needed). It comes down to formulating an integral equation that the solution has to obey and from it constructing a sequence of successive approximations, the convergence of which follows readily from the Lipschitz condition. Chapter six concludes by covering the existence and uniqueness of solutions to systems and n-th order equations.
About the approximately three hundred homework exercises strewn liberally throughout the text: many problems turn out not to be especially hard, they are often arranged in sequence so that having solved the earlier ones one knows what to do for the later. To proceed with dispatch, use Mathematica to perform the integrations or find roots of indicial polynomials or carry out substitutions for you (since this aspect of the derivation is not of the essence). For it comes down to one’s pedagogical philosophy: if one be confident in his grasp of the calculus little purpose would be served by seeking to work out the solutions without the aid of software. The hints can be almost too complete and thereby deprive one of a sense of accomplishment in figuring out on one’s own what to do (they basically sketch the solution for you). Nevertheless, though the problems are often straightforward they do illustrate concepts it is important to know (for instance, a problem on partial fraction decompositions). Thus, one should attempt all the problems. If routine, they help to internalize what one has just read; if perplexing (about one in every ten), they force one to think harder for an hour or two and thereby afford an ideal learning opportunity. But none remotely approaches the legendary difficulty of many of the problems in baby Rudin (for our review of this classic textbook in introductory analysis, see here).
The treatment in the present work leans to the theoretical, with the consequence that the subject begins to seem orderly, more so than anticipated in view of its reputation as but a grab bag of techniques lacking full generality (as say Courant and John portray it in the second volume of their course on calculus, our review here). Though often billed as being on the advanced undergraduate level, the present text is not perhaps quite so difficult. Everything in the exposition follows logically without too much effort, all of the problems that could pose a challenge are supplied with generous hints anyway and only occasionally do they call on tools from introductory analysis (to show convergence of a series) – raising the danger that the careless student might be lulled into a sense of complacency: for graduate-level textbooks certainly will expect the student to bring more of his own wit to the table and to display initiative.
Recommended for the serious student of pure mathematics looking for a competent guide into the subject of ordinary differential equations, without wanting to be detained by an overlong tour. For a more advanced, definitely graduate-level treatment see the standard reference by Coddington himself, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).
Displaying 1 - 2 of 2 reviews