Classes of second order nonlinear differential equations reducible to first order ones by variation of parameters

TL;DR

This paper extends the method of variation of parameters from linear to certain nonlinear second order differential equations, enabling their reduction to first order equations, with classical examples demonstrated.

Contribution

It introduces a novel extension of the variation of parameters method to nonlinear second order equations, facilitating their reduction to simpler forms.

Findings

Classical equations like Bernoulli, Riccati, and Abel are recovered.

The method successfully reduces specific nonlinear equations to first order.

Illustrated examples demonstrate the practical application of the extended method.

Abstract

The method of parameter variation for linear differential equations is extended to classes of second order nonlinear differential equations. This allows to reduce the latter to first order differential equations. Known classical equations such as the Bernoulli, Riccati and Abel equations are recovered in illustrated relevant examples.