= Solution
For $l=0$, put $u(r)=rR_0(r)$. The radial <Schrödinger equation> reduces to $u''+k'^2u=0$ inside and $u''+k^2u=0$ outside, where $k'^2=2m(E+V_0)/\hbar^2$ and $k^2=2mE/\hbar^2$. Regularity at the origin requires $u(0)=0$, excluding the inner cosine. A real outer sine-cosine combination can be written with an amplitude and a <phase shift>:
$$
\boxed{R_0(r)=
\begin{cases}
A\,\sin(k'r)/r,&0r\le a,\\
B\,\sin(kr+\delta_0)/r,&ra.
\end{cases}}
$$
The inner expression has the finite limiting value $Ak'$ at the origin. An overall complex normalization can carry the asymptotic phase convention of part a.
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