Ordinary Differential Equations

Abstract

The problematics of ordinary differential equations is very broad. Therefore, it was not possible to cover the whole field, in our Survey. For example, the reader will not find topics here, concerning theory of transformations of differential equations, optimal control, abstract differential equations, etc. The book being written for a very wide circle of readers, we tried, as far as possible, to work with classical concepts and methods (we do not consider solutions in the Kurzweil sense, for example). Systems of equations are not treated in the vector form exclusively, because for many readers the classical treating in components is more usual. For similar reasons it was not possible, when considering nonlinear equations, to avoid the concepts of general, or singular integrals which are quite current in technical literature. At the same time, we are well aware of the fact that these concepts are rather vague and thus not often used in modern mathematical literature. The encyclopedical character of the book made in necessary to include, in §§17.5 and 17.7, methods for solving equations of some special types. However, the reader’s attention should be drawn to the fact that numerical methods, discussed in Chap. 25, represent often a more effective tool of their solution.