AbstractThe Taylor series method is one of the earliest analytic‐numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of these algorithms is based on the approximate calculation of higher order derivatives using well‐known technique for the partial differential equations. The implicit extension based on a collocation term added to the explicit truncated Taylor series. This idea is different from the general collocation method construction, which led to the implicit R‐K algorithms [1].