The newly proposed analytic approximate solution method in the recent publications [Turkyilmazoglu, M.  "Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type," Appl. Math. Mod. 37, 7539-7548; Turkyilmazoglu, M.  " An effective approach for numerical solutions of high-order Fredholm integro-differential equations," Appl. Math. Comput. 227, 384-398; Turkyilmazoglu, M.  " Parabolic partial differential equations with nonlocal initial and boundary values," Int. J. Comput. Methods, doi: 10.1142/S0219876215500243] is extended in this paper to solve initial and boundary value problems governed by any order linear differential equations whose exact solutions are hard to obtain. Exact solutions are found from the method when the solutions are themselves polynomials. Better accuracies are achieved within the method by increasing the number of polynomials. Comparisons with some available methods show the ability of the proposed technique, even performing much better than the traditional Taylor series expansion.