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In this article, we presented a non-uniform mesh size high order exponential finite difference scheme for the numerical solutions of two point boundary value problems with  Dirichlet's boundary conditions. Under appropriate conditions, we have discussed the local truncation error and the convergence of the proposed method. Numerical experiments have been carried out to demonstrate the use and high order computational efficiency of the present method in several model problems. Numerical results showed that the proposed method is accurate and convergent. The order of accuracy is at least cubic which is in good agreement with the theoretically established order of the method.