The nucleation and growth of surface islands in the pre-coalescence stage has previously been studied by different methods, including the rate equation approach and kinetic Monte Carlo simulations. However, full understanding of island growth kinetics and the scaling properties of their size distributions is still lacking. Here, we investigate rate equations for the irreversible homogeneous growth of islands in the continuum limit, and derive a general island-size distribution whose shape is fully determined by the dynamics of the monomer concentration at a given size dependence of the capture coefficients. We show that the island-size distribution acquires the Family-Viscek scaling shape in the large time limit if the capture coefficients are linear in size for large enough islands. We obtain analytic solutions for the time-dependent monomer concentration, island density, average size and island-size distribution, which are valid for all times, and the analytic scaling function in the large time limit. These results can be used for modeling growth kinetics in a wide range of systems and shed more light on the general properties of the size distributions of different nano-objects.