An implicit numerical spline method for systems for ODEs

An implicit numerical method is introduced that gives piecewise polynomial spline approximations for the solution of an initial value problem for the equationy^'(t)=f(t,y(t)).Here, f is a function having continuous derivatives up to the order r. The method presented in this work, provides spline functions which approximate the solution of the problem in the most regular function space C^r^+^1. Error estimates in C and C^r^+^1 are given and the stability of the method is investigated.