A matrix method, which is called Taylor-matrix method, for the approximate solution of linear differential equations with specified associated conditions in terms of Taylor polynomials about any point. The method is based on, first, taking truncated Taylor series of the functions in equation and then substituting their matrix forms into the given equation. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. To illustrate the method, it is applied to certain linear differential equations and the generalized Hermite, Laguerre, Legendre and Chebyshev equations given by Costa and Levine [Int. J. Math. Edu. Sci. Technol. 20 (1989) 1]. (C) 2002 Published by Elsevier Science Inc.