Аннотация:This is a short note which summarizes several facts concerning the explicit solutions of linear systems of ordinary differential equations. The system x`=A(t)x, x(t0)=x0 has a simple solution — in terms of a matrix exponential — if A(t) and its antiderivative commute over the interval in which the solution of the system is thought. If they don’t commute, an integral equation can be written for the solution and an algorithm — whose complexity is close to that of the algorithm for computation of an exponential function — derived for its estimation; this is specially useful when the matrix and its antiderivative “almost commute”. A similar analysis is made for the nonhomogeneous case, as well as for an eigenvalue problem.