A Higher Order Non--Linear Differential Equation and a Generalization of the Airy Function

TL;DR

This paper introduces a higher order nonlinear differential equation that generalizes the Airy function through explicit solutions, expanding the mathematical understanding of such equations and their transformations.

Contribution

It presents a new higher order nonlinear differential equation and constructs explicit solutions, generalizing the classical Airy function via the Cole-Hopf transformation.

Findings

Explicit solutions for the even case are constructed.

The equation generalizes the classical Airy function.

The Cole-Hopf transformation links the nonlinear equation to a higher order Airy equation.

Abstract

In this paper a higher order non-linear differential equation is given and it becomes a higher order Airy equation (in our terminology) under the Cole-Hopf transformation. For the even case a solution is explicitly constructed, which is a generalization of the Airy function.