A numerical method based oil Fluctuationlessness Approximation. which wits developed recently, is constructed for solving Boundary Value Problems of Ordinary Differential Equations on appropriately defined Hilbert Spaces. The numerical solution is written in the form of it Maclaurin series. The unknown coefficients of this series are determined by constructing an (n - 2) unknown containing linear system of equations The eigenvalues of the independent variable's matrix representation are used in the construction of the matrices and the vectors of the linear system. The numerical Solution obtained by Fluctuationlessness Approximation is then compared with the Maclaurin coefficients of the analytical solution to observe the quality of the convergence Sonic illustrative examples are presented in order to give all idea about the efficiency of the method explained here.