Differential equation for the Uehling potential

The second-order differential equation for the Uehling potential is derived explicitly. The right side of this differential equation is a linear combination of the two Macdonald's functions $K_{0}(b r)$ and $K_{1}(b r)$. This central potential is of great interest in many QED problems, since it describes the lowest-order correction for vacuum polarization in few- and many-electron atoms, ions, muonic and bi-muonic atoms/ions as well as in other similar systems.