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Ordinary differential equations. Introduction to the theory of ordinary differential equations in the real domain. Transl. from the Czech by Michal Basch. (English) Zbl 0667.34002
Studies in Applied Mechanics, 13. Amsterdam etc.: Elsevier; Prague: SNTL Publishers of Technical Literature. 440 p. (1986).
The purpose of this book, translated from a Czechoslovak edition, is to serve as an introduction to the theory of ordinary differential equations and, for a lesser part, as a presentation of results obtained in Czechoslovakia since 1950. The concept of primitive function, and its properties, are recalled in a preliminary part, that of phase trajectory in chapter 1. Chapter 2 is devoted to elementary methods of integration, with many significant examples. Vectorial notation is introduced in chapter 3, and in particular used in the following chapter for linear equations of high order. The other chapters concern classical questions such as: how to solve autonomous linear equations, periodic linear ones; asymptotic behaviour of solutions, linear boundary value problems for differential operators in a compact interval, existence and uniqueness of solutions, global properties, differentiability of solutions with respect to initial conditions, their continuous dependance with respect to parameters, in the nonlinear case. A chapter is devoted to the Carathéodory theory, with application to the Filippov problem (equation the right-hand side of which is discontinuous). For the most part of the book, chapters deal in parallel with cases where the dependant variables are either real or complex. The book concern pure mathematicians. Indeed, in spite of parts related to applications of the dynamic systems theory (Van der Pol equation, relaxation oscillations, Filippov problem,...), the matter presentation is not directly useful for applied mathematicians, researchers, engineers working on dynamical systems of the various scientific disciplines, the classical books of the Andronov’ School (for example) giving a more workable information for applications.
Reviewer: C.Mira

34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34Axx General theory for ordinary differential equations
34Bxx Boundary value problems for ordinary differential equations
94Cxx Circuits, networks
34Dxx Stability theory for ordinary differential equations