Abstract
In this paper, we implement the shifted Jacobi operational matrix of derivative with spectral tau method and collocation method for numerical solution for the systems of linear and non-linear ordinary differential equations subject to initial or boundary conditions. By means of this approach, such problems are reduced for solving a system of algebraic equations and are greatly simplified the problems. We compare the obtained numerical results with the exact solutions. Also, we present and prove several theorems, which are related to the convergence of the proposed methods. Finally, some numerical test examples are presented to illustrate the validity and the great potential of the proposed technique.