We consider the problem of computing the Liapunov and the period constants for a smooth differential equation with a non degenerate critical point. First, we investigate the structure of both constants when they are regarded as polynomials on the coefficients of the differential equation. Secondly, we take advantadge of this structure to derive a method to obtain the explicit expression of the above-mentioned constants. Although this method is based on the use of the Runge-Kutta-Fehlberg methods of orders 7 and 8 and the use of Richardson's extrapolation, it provides the real expression for these constants.