Differential equations. An introduction to basic concepts, results and applications. (English) Zbl 1070.34001

Vrabie, Ioan I.

River Edge, NJ: World Scientific (ISBN 981-238-838-9/hbk). xvi, 401 p. (2004).

This nice text is a completely rewritten English version of the book which has been published first in Romanian by the author [Differential equations (Ecuatii diferentiale). Bucuresti, Matrix Rom (1999; Zbl 0994.34001)] and springs up from the lecture notes for the course in differential equations taught by the author in the Faculty of Mathematics at the University of Iaşi during the past twelve years. The principal idea of the author to teach differential equations with main emphasis on the initial value problems (termed in the book Cauchy problems) determines the way the material is presented. The book comprises eight chapters.
Chapter one starts with a very nice historical introduction to the subject describing contributions to differential equations from Leibniz, Newton, Bernoulli to the state-of-the-art research in the twentieth century. Then basic methods for the integration of fundamental classes of equations (linear, homogeneous, Bernoulli, exact, etc.) are explained and a variety of problems which lead to differential equations (radioactive decay, carbon dating method, mathematical pendulum, predator-prey model, etc.) are reviewed. Chapter two is one of the principal in the book and deals with the fundamental properties of solutions. The author addresses local existence of solutions, uniqueness of solutions, characterization of continuable solutions, types of saturated solutions, continuous dependence of solutions on initial data and parameters, differentiability of solutions with respect to initial data and parameters, and initial value problems for \(n\)th-order equations. An in-depth discussion of theoretical methods for approximating solutions of differential equations is carried out in Chapter three, where classic power series method, Picard’s method of successive approximations and Euler’s method of polygons, as well as a new approach for theoretical approximation of solutions similar to Euler’s method of polygons are presented. Chapter four deals with systems of linear differential equations and \(n\)th-order linear differential equations. Here, the exponential of a matrix and a method for its computation are presented. Fundamentals of the stability theory form the subject of Chapter five, where different types of stability are introduced and studied. Stability of dissipative systems, controlled systems and chaos are briefly discussed. First integrals of autonomous and nonautonomous ordinary differential equations, first integrals of first order partial differential equations, Cauchy problem for quasilinear partial differential equations and properties of conservation laws are studied in Chapter six. Further extensions of the concept of ordinary differential equation are considered in Chapter seven, where advanced material related to generalized solutions, Carathéodory differential equations, differential inclusions, variational inequalities, necessary and sufficient conditions for invariance, and gradient systems is collected. Finally, the last chapter contains auxiliary results from vector analysis, Ascoli-Arzelá theorem and related facts on compactness for continuous vector-functions, and introduces projection of a point on a convex set. Each chapter except the last one concludes with a nice selection of exercises and problems whose level varies from simple to challenging. Remarkably, complete solutions to most problems are provided at the end of the book. The bibliography contains over eighty items including many well-known references on differential equations and a number of textbooks in Romanian. Although the English could be corrected to improve the presentation, this book is a nice contribution to existing literature on differential equations and can be warmly recommended as additional reading for students in mathematics, physics and engineering who need solid theoretical background in differential equations.

Reviewer: Yuri V. Rogovchenko (Famagusta)

MSC:

34-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34A05	Explicit solutions, first integrals of ordinary differential equations
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A30	Linear ordinary differential equations and systems
34D20	Stability of solutions to ordinary differential equations
35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations

Keywords:

ordinary differential equations; integration techniques, existence; uniqueness; continuous dependence on initial data and parameters; saturated solutions; stability; differential equations with discontinuous right-hand sides; first integrals

Citations:

Zbl 0994.34001