ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH NONLINEARITIES, THAT ARE COMPOSITIONS OF EXPONENTIAL AND REGULARLY VARYING FUNCTIOS

Abstract

One of the most actual problems of the modern qualitative theory of ordinary differential equations is the study of nonlinear and, especially, significantly nonlinear non-autonomous differential equations. Among the works in this area related to establishing the asymptotic properties of solutions, the largest part consists of studies of equations with power-law nonlinearities and nonlinearities asymptotically close to power-law nonlinearities, as well as with exponential nonlinearities. The premise of these studies was the study of the Emden–Fowler equation, partial cases of which are used in nuclear physics, gas dynamics, fluid mechanics, relativistic mechanics, and other fields of natural science. The existence conditions and asymptotic representations of a sufficiently wide class of solutions of substantially nonlinear second-order differential equations are found in the paper. This class of solutions was introduced in the works of V. M. Evtukhov for equations of the Emden-Fowler type of the nth order and specified for the equation of the second order. The investigated differential equations contain nonlinearities, which are compositions of exponential and correctly variable when the argument is directed to a special point of the functions. An important difference of this class of equations is the impossibility of even asymptotically representing the nonlinearity in the form of a product of functions, each of which depended either only on the unknown function or only on the derivative of the unknown function. The class of studied solutions contains properly variable solutions of such equations. In the work, asymptotic images are obtained both for the solutions of the studied class and for their first-order derivatives.