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A separating surface for the Painlevé differential equation x '' =x 2 -t. (English) Zbl 0835.34035

The author is concerned with the Painlevé equation (1) x '' =x 2 -t. The equation (1) has exactly three different types of solutions for t>0: (A) a 2-parameter family of solutions that oscillate infinitely often and slowly approach -t as t; (B) a 1- parameter family of solutions asymptotic to t as t; and (C) a family of solutions, each of which tends to + at some finite value of t.

It is shown that the solutions of type (B) form a surface 𝕊 3 separating the solutions of type (A) from these of type (C). Every solution to type (B) must enter the interior of the parabola 𝒫={(t,x):x 2 -t=0} for the final time through one of the points P τ :(t,x)=(|τ|,sign(τ)|τ|) with positive and uniquely determined slope a * (τ), where τ is a real parameter. On the other hand, for each point P τ on 𝒫 there exists a unique positive slope a * (τ) such that the solution x a * (t) of (1) throughout P τ with the slope a * is asymptotic to +t as t. The function a * is proved to be continuous and differentiable everywhere except at τ=0. Several approximate values of a * (τ) and the graphs of a * (τ) and of the surface 𝕊 are given.

Reviewer: J.Kalas (Brno)
MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory