The frequency-independent source function for a two-level atom in a self-excited atmosphere with no continuous absorption is encountered, for example, in theories of line formation in the solar corona and outer chromosphere. We present accurate numerical solutions to this transfer problem in finite and semi-infinite atmospheres for physically interesting values of the relevant parameters and discuss the mathematical and physical features of the source functions we obtain. We do not here apply our results to any particular astrophysical problem, but rather concentrate on understanding the physics of the transfer problem. Particular attention is given to the effects of collision broadening. Finally, the assumption of a frequency-independent source function for Doppler broadening is examined for self-consistency.