First integrals and reduction of a class of nonlinear higher order ordinary differential equations. (Q1417980)

This paper is concerned with the resolution of nonlinear higher-order equations of the following form \[ x^{(n)}= h(x, x^{(n-1)}\dot x),\text{ where }\dot x= {dx\over dt}\text{ and }x^{(n)}= {d\over dt} (x^{(n-1)}),\;n\in \mathbb{N}^*. \] The proposed method consists in constructing their first integrals and reducing them. The cases \(n= 3,4\), and \(5\) are discussed in detail and some examples are given.