ASYMPTOTIC REPRESENTATIONS OF SOLUTIONS WITH SLOWLY VARYING DERIVATIVES OF THE SECOND ORDER DIFFERENTIAL EQUATIONS WITH THE PRODUCT OF DIFFERENT TYPES OF NONLINEARITIES

Abstract

Signi cantly nonlinear non-autonomous di erential equations have begun to appear in practice from the second half of the nineteenth century in the study of real physical processes in atomic and nuclear physics, and also in astrophysics. The di erential equation, that contains in its right part the product of regularly and rapidly varying nonlinearities of an unknown function and its rst-order derivative is considered in the paper. Partial cases of such equations arise, rst of all, in the theory of combustion and in the theory of plasma. The rst important results on the asymptotic behavior of solutions of such equations have been obtained for a second-order di erential equation, that contains the product of power and exponential nonlinearities in its right part. For, no such equations have been obtained before. According to this, the study of the asymptotic behavior of solutions of nonlinear di erential equations of the second order of general case, that contain the product of regularly and rapidly varying nonlinearities as the argument tends either to zero or to in nity, is actual not only from the theoretical but also from the practical point of view. The asymptotic representations, as well as the necessary and su cient conditions of the existence of Pω(Y0, Y1,±∞)-solutions of such equations are investigated in the paper. This class of solutions is the one of the most di cult of studying due to the fact that, by the a priori properties of the functions of the class, their second-order derivatives aren't explicitly expressed through the rst-order derivative. The results obtained in this article supplement the previously obtained results for Pω(Y0, Y1,±∞)-solutions of the investigated equation concerning the su cient conditions of their existence and quantity.