= Solution
For a <radial function>, the <Laplacian> is $\Delta\phi=\phi_{rr}+2\phi_r/r$. Introduce the <retarded and advanced null coordinates> and the <radial reduction of the three-dimensional wave equation>:
$$
u=t-r,\qquad v=t+r,\qquad r=\frac{v-u}{2},\qquad w=r\phi.
$$
The <chain rule> gives $\partial_t=\partial_u+\partial_v$ and $\partial_r=-\partial_u+\partial_v$. Consequently the inhomogeneous <wave equation> is
$$
\boxed{-4\phi_{uv}+\frac{4}{v-u}(\phi_v-\phi_u)=F.}
$$
Multiplying the radial <wave equation> by $r$ removes its first-order radial derivative:
$$
-w_{tt}+w_{rr}=rF.
$$
Thus the particularly useful <retarded and advanced null coordinates> form is
$$
\boxed{-4w_{uv}=rF,\qquad
\partial_u\partial_v\bigl((v-u)\phi\bigr)=-\frac{v-u}{4}F.}
$$
These formulas hold for $r>0$. At the axis, smooth <spherical symmetry> imposes $\phi_r(t,0)=0$ and $w(t,0)=0$; equivalently $w$ has a smooth <odd extension> across $r=0$.
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