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Asymptotic behavior of the general real solution of a Painleve equation of the third kind. (English) Zbl 0674.34048

The author here considers the equations

$\left(1\right)\phantom{\rule{1.em}{0ex}}{u}^{\text{'}\text{'}}+{x}^{-1}{u}^{\text{'}}+sinu=0$

and gives a complete description of the asymptotic behaviour of the real solutions of (1) with the initial condition $u\left(x\right)=rlnx+s+O\left({x}^{2}\right)$ as $x\to 0$. A special class of complex solutions of (1) is also identified which have no singularities for real x and which become solutions of the Hankel type

$u\sim C{H}_{0}\left(x\right)\to \alpha {x}^{-1/2}exp±i\left(x+\beta \right)$

at infinity.

Reviewer: J.O.C.Ezeilo
##### MSC:
 3.4e+06 Asymptotic expansions (ODE)
##### Keywords:
second order differential equation; Hankel solution