  The main purpose of this work is to introduce and analyse some
generalizations of diverse superposition rules for first-order differential
equations to the setting of second-order differential equations. As a result,
we find a way to apply the theories of Lie and quasi-Lie systems to analyse
second-order differential equations. In order to illustrate our results,
several second-order differential equations appearing in the physics and
mathematical literature are analysed and some superposition rules for these
equations are derived by means of our methods.
