Majorana transformation for differential equations

TL;DR

This paper introduces a method to reduce the order and non-linearity of certain scale-invariant differential equations, enabling transformations like converting second-order Emden-Fowler equations into first-order Abel equations, generalizing Majorana's approach.

Contribution

It develops a generalized method for simplifying scale-invariant differential equations and reducing their non-linearity, extending Majorana's original technique.

Findings

Transformations of Emden-Fowler equations into Abel equations

Reduction of differential equation order and non-linearity

Generalization of Majorana's method for broader applications

Abstract

We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of non-linearity of the lower-order equation. Some applications of these methods are carried out and, in particular, we show that second-order Emden-Fowler equations can be transformed into first-order Abel equations. The work presented here is a generalization of a method used by Majorana in order to solve the Thomas-Fermi equation.