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A separating surface for the Painlevé differential equation ${x}^{\text{'}\text{'}}={x}^{2}-t$. (English) Zbl 0835.34035

The author is concerned with the Painlevé equation (1) ${x}^{\text{'}\text{'}}={x}^{2}-t$. 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 $-\sqrt{t}$ as $t\to \infty$; (B) a 1- parameter family of solutions asymptotic to $\sqrt{t}$ as $t\to \infty$; and (C) a family of solutions, each of which tends to $+\infty$ at some finite value of $t$.

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

Reviewer: J.Kalas (Brno)
##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory