Master generators: A novel approach to construct and solve ordinary differential equations

Abstract

Ordinary differential equations (ODEs) play a crucial role in applied mathematics, engineering, and various other fields. With a rich history of study and countless applications, they have attracted the attention of researchers who have developed numerous methods for solving linear and non-linear ODEs. However, despite considerable progress, there remain many unsolved equations and challenges in finding appropriate solutions. This research introduces a groundbreaking method for generating an infinite number of generators for both linear and non-linear ODEs, based on the use of master improper integrals and a set of novel theorems. The proposed method enables the creation of infinite ODEs using a single generator, providing a versatile and powerful approach to tackling various types of differential equations. This research not only presents several specific generators derived from the theorems, but also establishes general forms for linear and non-linear generators, highlighting their adaptability to a wide range of scenarios. In addition, we demonstrate the practical utility of the generators through real-world examples and applications across diverse fields such as quantum field theory, astrophysics, and fluid dynamics. These examples showcase the potential of the proposed method in solving complex problems and its far-reaching impact on multiple disciplines. Through a comprehensive presentation of the theoretical underpinnings, derivation of generators, and application examples, this research aims to provide a thorough understanding of the infinite generators for linear and non-linear ODEs, as well as to inspire further research and development in this fascinating and vital area of applied mathematics. The introduction of infinite generators not only addresses the existing challenges in solving ODEs but also opens up new avenues for exploring novel solutions and approaches, ultimately contributing to the advancement of various fields that rely on differential equations.