This paper presents a new method based on Fluctuationlessness Theorem that was proven recently for getting the numerical solutions to the Ordinary Differential Equations over appropriately defined Hilbert Spaces. Approximations to the solution are evaluated at a series of discrete points. These points are constructed as the eigenvalues of the independent variable's matrix representation restricted to an n dimensional subspace of the Hilbert Space under consideration. The approximated solution is written in the form of an n-th degree polynomial of the independent variable. The unknown coefficients are found by setting up a system of linear equations such that this solution satisfies the initial condition and the differential equation at the grid points. The numerical quality of the solution can be increased by taking greater values of n.