= Solution
With zero internal <pressure> and negligible <surface tension>, <pressure continuity> sets the liquid-side boundary value to $p(R,t)=0$. Evaluating the <pressure field of a pulsating sphere> there gives
$$
0=\frac1R(R^2\ddot R+2R\dot R^2)-\frac12\dot R^2+\frac{p_\infty}\rho.
$$
Thus the <Rayleigh equation for an inviscid spherical bubble>, together with its <initial conditions>, is
$$
\boxed{R\ddot R+\frac32\dot R^2=-\frac{p_\infty}\rho,
\qquad R(0)=a,\quad\dot R(0)=0}.
$$
For $p_\infty>0$, initially $\ddot R(0)=-p_\infty/(\rho a)<0$, so the physically relevant branch describes <Rayleigh collapse of a spherical cavity>.
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