Abstract. We discuss the mth-order linear differential equation with matrix coefficients in terms of a particular matrix solution that enjoys properties similar to the exponential of first-order equations. A new formula for the exponential matrix is established with dynamical solutions related to the generalized Lucas polynomials.1. Introduction. The study of higher-order linear differential equations with square matrix coefficients is usually left aside for being equivalent to a first-order equation having the companion matrix as its coefficient. The handling of this matrix is not an amenable task for obtaining results proper of higher-order equations which are shadowed or simply unknown. The companion matrix, although sparse, does not preserve any common property that the involved matrix coefficients might share.In this paper, we shall present a direct study of higher-order equations in terms of a particular square matrix solution, which is characterized by zero initial "displacement" values and a unit initial "impulse" value, and that we shall refer to in the sequel as the dynamical solution. It will be shown that this solution enjoys intrinsic properties not necessarily shared by the complementary basis solutions, and that in a certain way the dynamical solution should do for higher-order equations what the

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A description of the fundamental solution of the mth-order linear ordinary differential equation with matrix coefficients is given in terms of power series and the Green function. The second-order equation is discussed.