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Ordinary Differential Equations in Theory and Practice

Robert Mattheij and Jaap Molenaar.
Publication Date: 
Number of Pages: 
Classics in Applied Mathematics
[Reviewed by
Henry Ricardo
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This book, a volume in SIAM's Classics in Applied Mathematics series, lives up to its title and fits somewhere between Ordinary Differential Equations by Carrier and Pearson and the second edition of Ordinary Differential Equations by Hartman, also "classics" in the series. This no-nonsense book is less heuristic than the Carrier/Pearson text, yet not as theoretical as Hartman's treatise. Originally published in 1996 by Wiley, the book is a successful effort to present analytical and numerical techniques as two sides of the same coin, as well as to provide an introduction to mathematical modeling. The text's main source of examples is classical mechanics.

Typos and errors in the original edition have been corrected, with no major changes in the text itself. In the prefaces to the original and current editions, the authors have provided Internet addresses for numerical programs. The URLs are no longer valid, but a graduate student or researcher should be able to locate appropriate software. There are some strange typographical mutations on pp. 25-26, where norms are introduced. The references have not been updated for this edition. Also, there are some inaccuracies in the new Index (mostly, it seems, with respect to entries appearing toward the end of the book). For example, 'inverse resonance' appears on p. 320, not p. 321; a mention of the 'Maxwell equations' occurs on p. 303, not p. 305.

The book is intended for both undergraduate and graduate courses, but I don't believe it would be palatable to American undergraduates unless they have had multivariable calculus, linear algebra, and an introductory ODEs course. Laplace transforms and series solutions are missing, although there is a great emphasis on one-step and multistep numerical methods, as well as treatments of stability, chaotic systems, singular perturbations, and boundary value problems. There are many good examples (in smaller print), but they are stated briefly, with few details. There are well-chosen exercises, but relatively few and only at the end of each chapter, with no answers at the back of the book (as is common in European texts). Overall, this is a fine text for a graduate course in applied differential equations and a valuable reference that should be on the shelves of researchers and those teaching differential equations.

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

Preface to the Classics Edition; Preface; Chapter 1: Introduction; Chapter 2: Existence, Uniqueness, and Dependence on Parameters; Chapter 3: Numerical Analysis of One-Step Methods; Chapter 4: Linear Systems; Chapter 5: Stability; Chapter 6: Chaotic Systems; Chapter 7: Numerical Analysis of Multistep Methods; Chapter 8: Singular Perturbations and Stiff Differential Equations; Chapter 9: Differential-Algebraic Equations; Chapter 10: Boundary Value Problems; Chapter 11: Concepts from Classical Mechanics; Chapter 12: Mathematical Modelling; Appendices; References; Index.