This work aims to convert an eigenvalue problem for ordinary differential operators by reducing order and increasing the number of unknowns in such a way that the resulting equation is a first order vector differential equation, whose matrix structure is quite simple. This form enables us to develop a factorization scheme around the zero value of the independent variable. The scheme presents an infinite product where finite truncations can be used as approximants for the solution. This work considers these points and discusses the determination of the eigenvalues. (c) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.