*(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 $\mathbb{S}\subset {\mathbb{R}}^{3}$ separating the solutions of type (A) from these of type (C). Every solution to type (B) must enter the interior of the parabola $\mathcal{P}=\{(t,x):{x}^{2}-t=0\}$ for the final time through one of the points ${P}_{\tau}:(t,x)=\left(\right|\tau |,\text{sign}\left(\tau \right)\sqrt{\left|\tau \right|})$ 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 $\mathcal{P}$ 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 $\mathbb{S}$ are given.

##### MSC:

34C10 | Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory |