Asymptotic forms of solutions to the third Painlevé equation
The authors consider the implicit differential equation \[ -xyy''+ xy^{\prime 2}- yy'+ ay^3+ by+ cxy^4+ dx= 0 \] which can be obtained from the third Painlevé equation by multiplying it by \(xy\). It is assumed that all complex parameters \(a\), \(b\), \(c\) and \(d\) are different from zero. By means of two- and three-dimensional power geometry [\textit{A. D. Bruno}, Power geometry in algebraic and differential equations. Transl. from the Russian. Corrected and enlarged ed. North-Holland Mathematical Library. 57. Amsterdam: North-Holland (2000; Zbl 0955.34002)], the authors determine near the singularities \(x= 0\) and \(x=\infty\) the asymptotic forms of the solutions.
scientific article; zbMATH DE number 5656097
scientific article