Asymptotic representations for the solutions of second-order differential equations with rapidly and regularly varying nonlinearities. (English. Ukrainian original) Zbl 1428.34071

J. Math. Sci., New York 238, No. 3, 316-332 (2019); translation from Neliniĭni Kolyvannya 20, No. 4, 549-563 (2017).
The author considers the equation \[ y''=\alpha_0p(t)\varphi_0(y)\varphi_1(y'), \] where \(\alpha_0=\pm 1\), \(p: [a,\omega)\to(0,\infty)\) with \(-\infty< a <\omega\le+\infty\), \(\varphi_1\) is a regularly varaing function, and the following limit holds \[ \lim_{y\to Y_0}\frac{\varphi_0(y)\varphi''_0(y)}{(\varphi'_0(y))^2}=1, \] in which \(Y_0\in\{0,\pm\infty\}\). Asymptotic representations for solutions are obtained. An illustrative example is given as well.


34D05 Asymptotic properties of solutions to ordinary differential equations
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
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