Qualitative analysis and series solutions for second-order ODEs: Frobenius and Floquet methods

• Grant number: 25/06697-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: January 01, 2026
• End date: December 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: João Henrique Santos de Andrade
• Grantee: Bruno Hernandes de Seixas
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/15567-8 - Qualitative properties for geometric differential equations, AP.R

Abstract

This undergraduate research project aims to study two classical and complementary techniques in the theory of ordinary differential equations (ODEs): the Frobenius method, which constructs power series solutions near regular singular points, and Floquet theory, which analyzes the stability of solutions to linear differential equations with periodic coefficients. These tools are fundamental in applied mathematics, physics, and dynamical systems. Based on the rigorous treatment by Coddington and Levinson (1984), the student will explore theoretical aspects, worked examples, and stability properties. The project also serves as a preparatory step for more advanced studies in partial differential equations (PDEs) and geometric analysis. (AU)