Starting from the Poynting theorem, which arises from the exact Maxwell equations, we establish the balance of energy for the radiating ideal two capacitors problem. This balance of energy results in a nonlinear differential equation governing the time evolution of the voltage V. Boykin, Hite and Singh, following an approach not based on first principles, were the first to obtain this nonlinear differential equation and proposed an exponentially decaying voltage as a unique solution for it. We claim that the space of solutions for this differential equation is much richer. In fact, besides the exponentially decaying solution just mentioned there exist solutions with a sudden death behavior. The radiative effect introduced by Boykin, Hite and Singh, complemented with our analysis based on the exact Maxwell equations and the characterization of the more general space of solution of the nonlinear differential equation, explain the missing energy paradox in the ideal two capacitors problem.