Riccati-parameter solutions of nonlinear second-order ODEs

TL;DR

This paper introduces a method to derive parametric solutions for certain nonlinear second-order ODEs using Riccati equations, simplifying the process of finding solutions in physically relevant problems.

Contribution

It demonstrates that a specific form of factorization combined with Riccati solutions yields a new class of parametric solutions for nonlinear second-order ODEs.

Findings

Parametric solutions can be systematically obtained using Riccati equations.

The method applies to many physically relevant nonlinear ODEs.

Riccati parameter acts as a growth parameter from trivial to particular solutions.

Abstract

It has been proven by Rosu and Cornejo-Perez in 2005 that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential operators. Here, it is shown that an interesting class of parametric solutions is easy to obtain if the proposed factorization has a particular form, which happily turns out to be the case in many problems of physical interest. The method that we exemplify with a few explicitly solved cases consists in using the general solution of the Riccati equation, which contributes with one parameter to this class of parametric solutions. For these nonlinear cases, the Riccati parameter serves as a `growth' parameter from the trivial null solution up to the particular solution found through the factorization procedure