non-linear differential equation, Cauchy problem, majorants method, a mobile neighborhood of a singular point, the approximate analytic solution.

The main task of the theory of differential equations is the existence and uniqueness of solutions. The peculiarity of nonlinear differential equations related to the presence of moving singular points, which relate to the class of such equations in the general case is not solvable in quadratures. It should be noted that the non-linear differential equations with moving singular points is not an analogue of the classical theorems of existence - Cauchy’s theorem, Picard theorem. In particular, the proof of Cauchy’s theorem is based on the method of majorant which is applied to the right side of the equation. This approach limits the ability to use the proof of this theorem to construct an analytical approximate solutions. In this paper we prove the existence and uniqueness of the solution of this class of nonlinear differential equations in the neighborhood of a singular point of the mobile used to the desired solution. This approach allows you to take advantage of the existence theorem for the construction of analytical approximate solutions.

Orlov Viktor Nikolaevich, Dr. Sci. Phys. & Math., Theory and Methods of Teaching Mathematics, Humanitarian and Pedagogical Academy (branch) of V. I. Vernadsky Crimean Federal University, Yalta, Russia; Professor at the Department of Mathematical Analysis, Algebra and Geometry of I. Y. Yakovlev Cheboksary State Pedagogical University, Cheboksary, Russia.

Ivanitskii Alexander Y., Candidate of Physical and Mathematical Sciences, Professor, Chuvash State University I. N. Ulyanov, Cheboksary, Russia.

Kudryashov Natalia V., Master 1st year, Chuvash State University I. N. Ulyanov, Cheboksary, Russia.