A class of degenerate parabolic equations of any order is considered. It is found a linear change of variables which reduces the equations from this class to degenerate parabolic equations of any order, which are natural generalizations of the classical Kolmogorov's equations of diffusion with inertia. Conditions on coefficients of the equations under which the offered change of variables is nondegenerate are established. By means of this change of variables the well-known result about the fundamental solution of the Cauchy problem for equations of the Kolmogorov type is disseminated on the considered class of the equations.