The task of finding the analytical solution of the nonlinear Poisson equation in the inversion layer of a metal-oxide-semiconductor for the most accurate Kingston-Neustadter (KN) model, of electrons and holes, has been a long-standing open problem of the semiconductor physics. We summarize here the new method that makes possible to solve the problem. We obtain the analytical solution for the KN model and compare its physical predictions with those of the numerous ad-libbed compact analytical models where simplified charge densities of only electrons or holes are assumed. As an example of the latter, we consider the Hauser-Littlejohn (HL) model. The charge distribution in the HL model, is closer to a 2D concentration, with a surface density that differs by orders of magnitude from the KN model. The source-drain currents in the inversion layer channel of junctionless (JL) MOSFETs differ also by orders of magnitude and depend on the impurity concentration.