Exact solution of the Landau-Lifshitz equations for a radiating charged particle in the Coulomb potential

We solve exactly the classical non-relativistic Landau-Lifshitz equations of motion for a charged particle moving in a Coulomb potential, including radiation damping. The general solution involves the Painlevè transcendent of type II. It confirms our physical intuition that a negatively charged classical particle will spiral into the nucleus, supporting the validity of the Landau-Lifshitz equation.