Approximations to an isolated solution of an mth order nonlinear ordinary differential equation with m linear side conditions are determined. These approximations are piecewise polynomial functions of order $m + k$ (degree less than $m + k$) possessing $m - 1$ continuous derivatives. They are determined by collocation, i.e., by the requirement that they satisfy the differential equation at k points in each subinterval, together with the m side conditions. If the solution of the sufficiently smooth differential equation problem has $m + 2k$ continuous derivatives and if the collocation points are the zeroes of the kth Legendre polynomial relative to each subinterval, then the global error in these approximations is $O(| \Delta |^{m + k} )$ with $| \Delta |$ the maximum subinterval length. Moreover, at the ends of each subinterval, the approximation and its first $m - 1$ derivatives are $O(| \Delta |^{2k} )$ accurate. The solution of the nonlinear collocation equations may itself be approximated by solving ...