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Differential Equations: An Introduction to Basic Concepts, Results and Applications

Ioan I Vrabie
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

This is the second edition of a text first published in 2003, now augmented with new chapters and new sections. The first edition of the book was an entirely rewritten English language version of lecture notes originally in Romanian. The goal of the book is to present a unified approach to differential equations, primarily ordinary differential equations, that emphasizes the continuous interplay between theory and applications.

The focus of about one-third of the book is on the initial value problem for ordinary differential equations (which the author consistently calls the “Cauchy problem”). The first chapter surveys the historical background, offers an introductory discussion of elementary equations amenable to solution by quadrature, and provides a summary of common mathematical models based on differential equations. After this the author presents a rigorous exploration of fundamental results for the initial value problem: existence, uniqueness, continuation of solutions, dependence on parameters and approximation methods. The two chapters that follow include mostly standard material on systems of linear ordinary differential equations and stability analysis.

Then there are the less standard topics. The first of these the author calls “prime integrals”; these are quantities that are constant along solution curves, sometimes called constants of the motion. Beginning with results for autonomous and non-autonomous first order ordinary differential equations the author proceeds to first order linear and quasi-linear partial differential equations, including existence and uniqueness results as well as conservation laws.

A kind of catch-all chapter called “Extensions and Generalizations” discusses distribution solutions, Carathéodory solutions, differential inclusions and half a dozen other special topics. Items that have been added to this second edition follow. These are chapters on Volterra equations (briefly introduced) and the calculus of variations (also relatively short, focusing on fundamental results) as well as a section on the Laplace transform.

Each chapter has a collection of exercises, which are more routine, and problems, which usually require a proof or more complex calculation. Solutions to all exercises and problems are included.

This text would be suitable for courses at the beginning graduate level. It has a somewhat eccentric collection of topics, and does not include some of the more standard material on the qualitative theory now often taught in such courses. The language of the book is also something of a handicap. Word usage and syntax are occasionally awkward, and there are sentences here and there that are impenetrable. Better editing would be highly beneficial.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

  • Generalities
  • The Cauchy Problem
  • Approximation Methods
  • Systems of Linear Differential Equations
  • Elements of Stability
  • Prime Integrals
  • Extensions and Generalizations
  • Volterra Equations
  • Calculus of Variations