Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 4 / f / Solution

At the source, $\zeta =z_0={\mathcal{B}}_0/\beta$. The initial width required to lie exactly on the similarity solution is thereforeThe corresponding $Q_0=A{z}_0^{4/3}$ and $M_0=C{z}_0^{5/3}$ are the compatible initial fluxes stipulated in the question.