Asymptotic representations of the Pω(Y0, Y1, ±∞)-solutions of the second order differential equation, which contains the product of different types of nonlinearities from an unknown function and its derivative

Abstract

The study of asymptotic representations of solutions of differential equations, in particular, of the second order differential equations, which contain nonlinearities of various types in the right-hand side, play an important role in the development of the qualitative theory of differential equations. This paper considers the type of differential equations of the second order, which contain in the right part the product of a regularly varying function from an unknown function and a rapidly varying function from the derivative of an unknown function when the corresponding arguments tend to zero or infinity. Necessary and sufficient conditions for the existence of slowly varying Pω(Y0, Y1, ±∞)-solutions of such equations have been obtained. Asymptotic representations of such solutions and their first-order derivatives are also obtained. Note that when additional conditions are imposed on the coefficients of the characteristic equation, such Pω(Y0, Y1, ±∞)-solutions of the equation exist as a one-parameter family. Similar results were obtained earlier when considering second-order equations, which contain in the right-hand side the product of a rapidly varying function from an unknown function and a regularly varying function from the derivative of an unknown function when the arguments tend to zero or infinity. For the equations considered in this paper, similar results are new.