# A new perspective on the Ermakov-Pinney and scalar wave equations

## TL;DR

This paper introduces a novel method linking Ermakov-Pinney and scalar wave equations to first-order non-linear equations using auxiliary 1-forms, with explicit applications in Kasner space-time.

## Contribution

It demonstrates that certain Ermakov-Pinney and scalar wave equations can be derived from first-order non-linear equations involving auxiliary 1-forms, providing new tools for physical applications.

## Key findings

- Explicit construction of auxiliary 1-forms in Kasner space-time
- Derivation of amplitude and phase functions for the scalar wave parametrix
- Potential applications in physical models using the new method

## Abstract

The first part of the paper proves that a subset of the general set of Ermakov-Pinney equations can be obtained by differentiation of a first-order non-linear differential equation. The second part of the paper proves that, similarly, the equation for the amplitude function for the parametrix of the scalar wave equation can be obtained by covariant differentiation of a first-order non-linear equation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary 1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi)=0, whereas the 1-form rho fulfills the non-linear equation div(rho)+rho**2=0. The auxiliary 1-forms (psi,rho) are evaluated explicitly in Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained. Thus, the novel method developed in this paper can be used with profit in physical applications.