In fluid mechanics, there is an essential deficiency of the analytical solutions of Navier-Stokes equations for 3D case of non-stationary flow. We explore the ansatz of derivation of non-stationary solution for the Navier-Stokes equations in the case of incompressible flow, which was suggested earlier. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard to the time-parameter t). But we find an elegant way to simplify it to the proper analytical presentation via exact solutions. The results are substantiated with illustrative examples, including numerical computations and appropriate figures. (C) 2016 Elsevier Ltd. All rights reserved.