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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