## Asymptotic representations of solutions with slowly varying derivatives of the second order differential equations with the product of different types of nonlinearities.(English)Zbl 1474.34353

Summary: Significantly nonlinear non-autonomous differential 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 differential equation, that contains in its right part the product of regularly and rapidly varying nonlinearities of an unknown function and its first-order derivative is considered in the paper. Partial cases of such equations arise, first of all, in the theory of combustion and in the theory of plasma. The first important results on the asymptotic behavior of solutions of such equations have been obtained for a second-order differential equation, that contains the product of power and exponential nonlinearities in its right part. In general case no such equations have been obtained before. According to this, the study of the asymptotic behavior of solutions of nonlinear differential 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 infinity, is actual not only from the theoretical but also from the practical point of view. The asymptotic representations, as well as the necessary and sufficient conditions of the existence of $$P_{\omega}(Y_0, Y_1, \pm\infty)$$-solutions of such equations are investigated in the paper. This class of solutions is one of the most difficult of studying due to the fact that, by the a priori properties of the functions of the class, their second-order derivatives are not explicitly expressed through the first-order derivative. The results obtained in this article supplement the previously obtained results for $$P_{\omega}(Y_0, Y_1, \pm\infty)$$-solutions of the investigated equation concerning the sufficient conditions of their existence and quantity.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 37C60 Nonautonomous smooth dynamical systems 26A12 Rate of growth of functions, orders of infinity, slowly varying functions
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