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The cylindrical KdV equation in this normalization is $q_t+q_{xxx}+q{q}_x+q/(3t)=0$, for $t\ne 0$. The additional time-dependent term distinguishes it from the ordinary KdV equation. A scaling-invariant solution has $q=t^{-2/3}F(x{t}^{-1/3})$. Substitution gives $F^{(3)}+F{F}^{\prime }-(\xi F{)}^{\prime }/3=0$. Integration reduces the similarity problem to $F^{\prime \prime }+F^2/2-\xi F/3=C$. The integration constant depends on supplementary conditions and cannot be set to zero without them. Use real cube roots separately on either sign of $t$.

Balance the time derivative, cubic dispersion and quadratic transport using the cylindrical KdV equation similarity ansatz $q(x,t)=t^{-2/3}F(\xi )$, $\xi =x{t}^{-1/3}$ on $t>0$. For $t<0$ the same real cube-root convention can be used separately. Substitution and removal of the common factor $t^{-5/3}$ giveEvery term is a total derivative: $F{F}^{\prime }=(F^2/2{)}^{\prime }$ and $\xi F^{\prime }+F=(\xi F{)}^{\prime }$. Integration therefore gives the requested second-order equationwhere $C$ is a constant determined by any additional boundary data. No condition in the question forces $C=0$.