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Asymptotics analysis of some bounded solution to the general third Painlevé equation. (English) Zbl 1141.34356

Summary: We study the general third Painlevé equation
\[ y''=\frac{y^{\prime 2}}{y}- \frac{y'}{x}+\frac1x\,(\alpha y^2+\beta)+\gamma y^3+\frac\delta y \]
where \(\alpha,\beta,\gamma\), and \(\delta\) are real parameters, discuss the boundedness of some solutions when \(\gamma < 0\) and \(\delta > 0\), and find an asymptotic representation of a group of oscillating solutions.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34D05 Asymptotic properties of solutions to ordinary differential equations
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