Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 31C / b / Solution

The scaling of the equation suggests $q=t^{-1/3}v(z)$, $z=x{t}^{-1/3}$, for $t>0$. Substitution givesIntegrating in $z$ gives the scaled Painlevé II equationwhere $C$ is an arbitrary integration constant. If the standard normalization $w^{\prime \prime }=2w^3+sw+\alpha$ is desired, set $z=3^{1/3}s$ and $v(z)=i{3}^{-1/3}w(s)$. Then $\alpha =-3iC$ andThe direct scaled equation avoids introducing imaginary variables when a real $v$ is required.