  We calculate the time-dependent formation of Bose--Einstein condensates
(BECs) in potassium vapours based on a previously derived exactly solvable
nonlinear boson diffusion equation (NBDE). Thermalization following a sudden
energy quench from an initial temperature $T_\mathrm{i}$ to a final temperature
$T_\mathrm{f}$ below the critical value and BEC formation are accounted for
using closed-form analytical solutions of the NBDE. The time-dependent
condensate fraction is compared with available $^{39}$K data for various
scattering lengths.
