Summary: A long-term aim of density functional theory is to obtain a differential equation for the ground-state electron density \(\rho(r)\) in a closed-shell atom, the simplest example being He. Since He remains intractable analytically, artificial atoms with harmonic repulsive potential energy \(u(r_{12})\) have therefore been studied. Here we exploit recent work on \(\rho(r)\) for such a two-electron system, with \(u(r_{12})= \lambda/r_{12}^2\), to construct a second-order linear differential equation for \(\rho(r)\). This is compared and contrasted with available results for different choices of \(u(r_{12})\).