nonlinear differential equation, Cauchy problem, method of majorant, a neighbourhood of a movable singular point, the existence theorem, analytic approximate solution, error estimate.

The theory of nonlinear differential equations associated with the name of the Italian mathematician Riccati and appeared in the early seventeenth century, continues to grow with new results to this day. A great contribution to the development of this theory has made mathematics of the Belarusian school, the results of research which are connected with the solvability of nonlinear differential equations in quadratures, but it can be done only in special cases [12]–[26]. The decision of some problems of mathematical theory of approximate analytical method of solving such equations are presented in [1]–[7]. The main cottage in the theory of differential equations is the existence theorem of solutions of differential equations. The peculiarity of nonlinear differential equations associated with moving singular points, which considerable reduces the significance of the classical theorems of existence. It should be noted that the classical existence theorem of the Cauchy theorem, theorem of Picard, not fully solve the problem for marked categories of differential equations. In this case, a variant of the method of majorants applied to the Cauchy theorem, limits the ability to use this theorem in the approximate method of solution. In this paper, given the existence proof of solutions of the nonlinear differential equation in a neighbourhood of a movable singular point in the new interpretation of the method of majorants. Built analytical approximate solution of the considered equation in a neighbourhood of a movable singular point in case the exact values of the latter.

1. Orlov Victor Nikolayevich e-mail: orlowvn@rambler.ru, Doctor of Physics and Mathematics, Head of the Department "Mathematics, theory and methods of teaching mathematics", Gumanitarno-pedagogical Academy (branch)" Vernadsky Crimean Federal center", Yalta, Russia; Doctor of Physics and Mathematics, Professor, I. Yakovlev Chuvash State Pedagogical University, Cheboksary, Russia. 2. Kolle Kseniya Valentinovna e-mail: ksenija-kolle2012@mail.ru, student of physics and mathematics, I. Yakovlev Chuvash State Pedagogical University, Cheboksary, Russia.