This paper is concerned with the construction of a new formula for integrating special second order, periodic and nonperiodic, initial value problems in ordinary differential equations. The presented formula is based on C 2-cubic splines as an approximation to the exact solution of the initial value problem, It turns out that the proposed method is a continuous extension of the well-known second order 1-stage implicit Nyström method and hence possesses nonvanishing intervals of periodicity and absolute stability.