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

Yıl 2018, Sayı: 2, 403 - 406, 19.08.2018

İrina Andreeva

Öz

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.

Anahtar Kelimeler

Differential equations, Ordinary solution, Special solution, Bifurcations

Kaynakça

• Andreev, A.F., & Andreeva, I.A. (2002). On a Question of Parametric Integration of Differential Equations. Vestnik St. Petersburg University: Ser.1. Mathematics, Mechanics, Astronomy, 4, 3- 10. Andreeva, I.A. (2003). Higher Mathematics. Special Solutions of Differential Equations of the First Order. St. Petersburg: SPbPU Publishing House. Andreev, A.F., & Andreeva, I.A. (2017). Investigation of a Family of Cubic Dynamic Systems. Vibroengineering Procedia, 15, 88 – 93. DOI: 10.21595/vp.2017.19389. Zalgaller, V.A. (1975). A Theory of Envelopes. Moscow: Nauka.