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A note on an open problem about the first Painlevé equation. (English) Zbl 1155.34372

From the introduction: We study the initial value problem of the first Painlevé equation:

y '' =6y 2 +x,y(0)=κ,y ' (y)=μ,

and find conditions on κ and μ for this problem to have an oscillating solution. This is an open problem posed by Peter A. Clarkson.

34M55Painlevé and other special equations; classification, hierarchies
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
[1]Clarkson, P.A. Painlevé equations-nonlinear special functions. www.uc3m.es/uc3m/dpto/MATEM summerschool/Leganes5.pdf
[2]Joshi, N., Kruskal, M.D. The Painlevé connection problem: an asymptotic approach I. Stud. Appl. Math., 86: 315–376 (1992)
[3]Qin, H.Z., Shang, N.N. Numerical analysis of the asymptotic solutions of the Painlevé equations. Numerical Solution and Computer Application, 26(1): 58–64 (2005)
[4]Qin, H.Z., Shang, N.N. Asymptotics analysis of a bounded solution to the general third Painlevé equation. MJMS, 18(2): 25–134 (2005)