A pair of coupled, nonlinear integro-differential equations, which exactly describe the transient current produced by the drift and collection of a swarm of space charge in a medium between two electrodes, has been derived for a single charge carrier species using a field-independent mobility model. The equations represent the solution of the mathematical problem involving arbitrary space-charge sizes and distributions coupled with the effects of the inherent time constants of the physical system. Computer-generated numerical solutions, for the special case of the initial charge distributions which correspond to steady-state currents, are reported. The equations under these conditions can be decoupled to form a stiff third-order differential equation. The numerical results indicate a possible scheme for correcting transient current waveforms, which are distorted by space-charge fields, in order to obtain the proper, ideal times of flight needed for true mobility measurements.