Superposition rules and second-order Riccati equations

Abstract

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose several generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest.

24 pages. Presentation improved and several relevant remarks added

References in corpus (3)

• Superposition rules, Lie theorem and partial differential equations
• A nonlinear superposition rule for solutions of the Milne--Pinney equation
• Integrability of Lie systems and some of its applications in physics

Cited by in corpus (1)

• A Quasi-Lie Schemes Approach to Second-Order Gambier Equations