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.

MSC:

 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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References:

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