Keywords

Algebra; Differential Equations; Differential Algebra

Abstract

Differential equations are a much-studied topic in the field of mathematics, as well as other sciences, such as engineering, economics, and biology. While much is known concerning these, there is still a large gap in our knowledge about such equations. It is important therefore, both to mathematics and other sciences, that we gain a more complete knowledge of differential equations, in particular the nature of their solutions. In this research, we investigate the solution space of linear ordinary differential equations (ODEs) from the standpoint of differential algebra. Differential algebra allows an ODE to be treated similarly to a polynomial, allowing analogous techniques such as factorization, and more largely, Galois Theory. In this work, we use differential algebraic techniques to study the factorization properties of second order ODEs. This in turn leads to an examination of the Riccati equations, a family of nonlinear ODEs, associated to those equations. While the connection between Riccati equations and linear ODEs has been studied previously, this work will offer a more complete analysis of this relationship. This will in turn be used to make conclusive statements regarding the solution space of the original ODEs. Moreover, we will present a new method for solving second order linear ODE’s, closely related to the method of D’Alembert Reduction.

Thesis Completion Year

2026

Thesis Completion Semester

Spring

Thesis Chair

Brennan, Joseph

College

College of Sciences

Department

School of Data, Mathematical and Statistical Sciences

Thesis Discipline

Mathematics

Language

English

Access Status

Open Access

Length of Campus Access

None

Campus Location

Orlando (Main) Campus

Additional Files
Lie-Galois Theory.docx (87 kB)
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