Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 16A / c / Solution

At the moving boundary,Substitution into the result of part (b) gives the Rayleigh collapse equationTreat $v=\dot{a}$ as a function of $a$, so that $\ddot{a}=vdv/da$. With $w=v^2$, the equation becomesMultiplication by the integrating factor $a^3$ after division by $a$ givesUsing $w(a_0)=0$,The collapsing branch has negative radial velocity, so