De integratione aequationum: an idea of Euler elaborated

Abstract Our aim is to revisit the fundamental theorem of higher-order homogeneous linear differential equations with constant coefficients in an elementary and direct manner. The approach is based on a recursive factorization of the attached differential operator and it is motivated by a comment of Euler. We use elementary arguments, like the Viéte formulae and the fundamental theorem of calculus, and also discuss, how this method applies to the nonhomogeneous case.