Rapidly varying solutions of general second-order differential equations that are close to differential equations with regularly varying nonlinearities are investigated. A crucial feature of such nonlinearities is that, in the general case, they cannot be represented as a product of functions, each depending on only one variable. This precludes the application of many methods developed for numerous generalizations of essentially nonlinear Emden–Fowler type differential equations, which were studied in detail in the works of R. G. Fowler, F. V. Atkinson, I. T. Kiguradze, T. A. Chanturia, N. A. Izobov, S. Belohorec, S. D. Taliaferro, J. S. W. Wong, O. V. Kostin, V. M. Evtukhov, and many of his students.

The study of rapidly varying solutions for equations of this type requires new approaches, as their asymptotic behavior in the neighborhood of a singular point differs substantially from the behavior of regularly varying functions, for which a much broader range of methods has been developed (see, for example, the monograph by V. Maric [7]).

In this paper, necessary and sufficient conditions are obtained for the existence of a class of rapidly varying solutions for the nonlinear differential equations under study. This class of solutions was introduced by V. M. Evtukhov for essentially nonlinear second-order differential equations based on a more general classification of solutions for nonlinear differential equations by I. T. Kiguradze. The properties of functions within this class allow for more detailed investigations into the asymptotic behavior of solutions and their derivatives. The existence of solutions with the identified asymptotic representations was proved by reducing the problem to the existence of solutions vanishing at the singular point for systems of quasilinear differential equations and applying the results of V. M. Evtukhov and A. M. Samoilenko for such systems. Precise asymptotic formulas have been obtained for this class of solutions and their first-order derivatives.

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