Logistic function as solution of many nonlinear differential equations

TL;DR

This paper demonstrates that the logistic function can serve as an exact solution to various nonlinear differential equations, including Riccati, second-order, and third-order types, and introduces a method for solving nonlinear PDEs using this function.

Contribution

It reveals the logistic function as a unifying solution for many nonlinear differential equations and proposes a simple method for solving nonlinear PDEs by comparison with standard equations.

Findings

Logistic function solves Riccati, second- and third-order nonlinear ODEs.

A new method for solving nonlinear PDEs using logistic function comparison.

Demonstrated wide applicability of logistic solutions in nonlinear differential equations.

Abstract

The logistic function is shown to be solution of the Riccati equation, some second-order nonlinear ordinary differential equations and many third-order nonlinear ordinary differential equations. The list of the differential equations having solution in the form of the logistic function is presented. The simple method of finding exact solutions of nonlinear partial differential equations (PDEs) is introduced. The essence of the method is based on comparison of nonlinear differential equations obtained from PDEs with standard differential equations having solution in the form of the logistic function. The wide application of the logistic function for finding exact solutions of nonlinear differential equations is demonstrated.