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 the language of Differential Equations. In particular, the classical theory of normal systems of Ordinary Differential Equations, supported by Cauchy theorems of existence and uniqueness of solutions, describes determined processes taking place in the Nature, technics and even in the society, i.e. such processes, for which a condition of a described system in an arbitrary fixed moment depends on its condition in any other moment. Solutions, describing such processes, are called the ordinary. But when the conditions of the Cauchy theorem are not satisfied, a situation totally changes. A point, in any neighborhood of which such conditions are not satisfied, may become for a system under consideration a point of non-uniqueness, a point of bifurcation. A solution of a system, each point of which appears to be a point of non-uniqueness, is called a special solution. A task of a full integration of a system demands finding of all its solutions, special solutions as well as ordinary ones. But this item shows us some gap in a special literature. This paper presents materials with the aim to fill this gap.

Keywords

• Differential equations,Ordinary solution,Special solution,Bifurcations

References

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