= Solution
Balance the time <derivative>, cubic dispersion and quadratic transport using the <cylindrical KdV equation> <similarity ansatz> $q(x,t)=t^{-2/3}F(\xi)$, $\xi=xt^{-1/3}$ on $t>0$. For $t<0$ the same real cube-root convention can be used separately. Substitution and removal of the common factor $t^{-5/3}$ give
$$
F^{(3)}+FF'-\frac13\xi F'-\frac13F=0.
$$
Every term is a total <derivative>: $FF'=(F^2/2)'$ and $\xi F'+F=(\xi F)'$. <Integration> therefore gives the requested second-order equation
$$
\boxed{F''+\frac12F^2-\frac13\xi F=C,}
$$
where $C$ is a constant determined by any additional boundary data. No condition in the question forces $C=0$.
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