We obtain analytical solutions of the non-linear Becker-Döring equations in open systems with input (deposition) and desorption of monomers, and size-linear forward and backward rate constants. We examine the cases of heterogeneous and homogeneous nucleation, where the cluster density is either constant or changes with time. In both cases, we find a governing parameter whose value determines the two different growth regimes with finite or infinite average size in the large time limit. In heterogeneous growth, exact analytical solution is given by the Polya distribution with a time-dependent average cluster size. In a more complex homogeneous growth, the equilibrium solution with a finite average size is also given by the Polya distribution. In the regimes with infinite average size, the Polya size distribution is established only if the cluster density tends to a constant. This requires particular time dependences of the attachment-detachment rate constants. We show that the obtained solution collapses to the analytic Family-Vicsek scaling function in the continuum limit.