Non-steady elastic waves diffraction on non-homogeneous transversally isotropic sphere is solved using surface transient functions general method. It is shown that the surface transient functions can be used to transform the problem for the inclusion to the initial-boundary value problem for first-order partial differential equations with boundary integral in time domain. To solve it a new first-order finite-difference method of Courant-Isakson-Riess type is proposed. The outer diffraction problem solution for the spherical non-homogeneous transversally isotropic inclusion is presented.