Notes to the Question of Presenting the Theme of Special Solutions of Ordinary Differential Equations in a University Course

Abstract

As Sir Isaac Newton has said, laws of the Nature have been written in thelanguage of Differential Equations. In particular, the classical theory ofnormal systems of Ordinary Differential Equations, supported by Cauchy theoremsof existence and uniqueness of solutions, describes determined processes takingplace in the Nature, technics and even in the society, i.e. such processes, forwhich a condition of a described system in an arbitrary fixed moment depends onits condition in any other moment. Solutions, describing such processes, are calledthe ordinary. But when the conditions of the Cauchy theorem are not satisfied,a situation totally changes. A point, in any neighborhood of which suchconditions are not satisfied, may become for a system under consideration apoint of non-uniqueness, a point of bifurcation. A solution of a system, eachpoint of which appears to be a point of non-uniqueness, is called a specialsolution. A task of a full integration of a system demands finding of all itssolutions, special solutions as well as ordinary ones. But this item shows ussome gap in a special literature. This paper presents materials with the aim tofill this gap.