# Dimensional analysis yields the general second-order differential equation underlying many natural phenomena: the mathematical properties of a phenomenon’s data plot then specify a unique differential equation for it

## TL;DR

This paper shows how dimensional analysis can derive a general second-order differential equation for various natural phenomena using only the mathematical properties of their data plots.

## Contribution

The novel approach uses dimensional analysis and data plot properties to derive differential equations without empirical constants.

## Key findings

- The general second-order differential equation is derived using dimensional analysis and data plot properties.
- The method successfully analyzes phenomena like the Standard Normal Distribution and Logistic Growth Function.
- The approach provides a unifying framework for understanding diverse phenomena through a common mathematical perspective.

## Abstract

This study uses dimensional analysis to derive the general second-order differential equation that underlies numerous physical and natural phenomena described by common mathematical functions. It eschews assumptions about empirical constants and mechanisms. It relies only on the data plot’s mathematical properties to provide the conditions and constraints needed to specify a second-order differential equation that is free of empirical constants for each phenomenon.

A practical example of each function is analyzed using the general form of the underlying differential equation and the observable unique mathematical properties of each data plot, including boundary conditions. This yields a differential equation that describes the relationship among the physical variables governing the phenomenon’s behavior. Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach.

It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.