This work focuses on the construction of a new method to solve the ordinary differential equations. The method presented here uses space extension concept. Main purpose of this work is to construct an approximation sheme to the solutions of the ordinary differential equations via two-term recursion formulae. In order to do this, the equation is converted to a new form which is called "Okubo Form" by introducing new unknowns into equation. Additionally, this work concerns with the series solution of the resulting Okubo Form and attempts to construct approximate solutions by truncating this series solution.