# §32.3 Graphics

## §32.3(i) First Painlevé Equation

Plots of solutions $w_{k}(x)$ of $\mbox{P}_{\mbox{\scriptsize I}}$ with $w_{k}(0)=0$ and $w_{k}^{\prime}(0)=k$ for various values of $k$, and the parabola $6w^{2}+x=0$. For analytical explanation see §32.11(i).

## §32.3(ii) Second Painlevé Equation with $\alpha=0$

Here $w_{k}(x)$ is the solution of $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$ and such that

 32.3.1 $w_{k}(x)\sim k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right),$ $x\to+\infty$;

compare §32.11(ii).

## §32.3(iii) Fourth Painlevé Equation with $\beta=0$

Here $u=u_{k}(x;\nu)$ is the solution of

 32.3.2 $\frac{{d}^{2}u}{{dx}^{2}}=3u^{5}+2xu^{3}+\left(\tfrac{1}{4}x^{2}-\nu-\tfrac{1}% {2}\right)u,$

such that

 32.3.3 $u\sim k\mathop{U\/}\nolimits\!\left(-\nu-\tfrac{1}{2},x\right),$ $x\to+\infty$.

The corresponding solution of $\mbox{P}_{\mbox{\scriptsize IV}}$ is given by

 32.3.4 $w(x)=2\sqrt{2}u_{k}^{2}(\sqrt{2}x,\nu),$

with $\beta=0$, $\alpha=2\nu+1$, and

 32.3.5 $w(x)\sim 2\sqrt{2}k^{2}{\mathop{U\/}\nolimits^{2}}\!\left(-\nu-\tfrac{1}{2},% \sqrt{2}x\right),$ $x\to+\infty$;

compare (32.2.11) and §32.11(v). If we set $\ifrac{{d}^{2}u}{{dx}^{2}}=0$ in (32.3.2) and solve for $u$, then

 32.3.6 $u^{2}=-\tfrac{1}{3}x\pm\tfrac{1}{6}\sqrt{x^{2}+12\nu+6}.$