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The concept of equivalent linearization and non-linearization for dynamic systems under Gaussian excitations with criteria in probability density space is considered in this paper. The term non-linearization used in the literature means the procedure of finding an equivalent nonlinear system for stochastically excited Hamiltonian system. New criteria of linearization and non-linearization and two approximate approaches are proposed. ID the first one, the direct minimization of a criterion is applied and the approximation of the probability density function by the truncated Grain-Charlier expansion is used. In the second approach, equivalent linearization or non-linearization is made for the Fokker-Planck equations corresponding to the original nonlinear and linearized or non-linearized dynamic systems, respectively. Two examples are given to illustrate the results obtained.