A Generalized Cole-Hopf Transformation for Nonlinear ODES

TL;DR

This paper introduces a hybrid Cole-Hopf-Darboux transformation that links solutions of nonlinear second order differential equations to linear ones, revealing connections to special functions of mathematical physics.

Contribution

It presents a new generalized transformation and a sufficient condition for relating nonlinear and linear second order differential equations.

Findings

Relates solutions of certain nonlinear equations to special functions.

Provides a sufficient condition for the transformation's validity.

Establishes a class of nonlinear equations connected to linear special functions.

Abstract

We introduce a hybrid Cole-Hopf-Darboux transformation to relate solutions of nonlinear and linear second order differential equations and derive a sufficient condition for this correspondence. In particular we show that solutions of some nonlinear second order equations are related to the special functions of mathematical physics through this transformation. These nonlinear equations can be viewed as the "class of special nonlinear equations" which correspond to the linear differential equations which define the special functions of mathematical physics.