Equações diferenciais ordinárias complexas com coeficientes constantes: solução geral e aplicações

Abstract

In this dissertation, we present the general theory of existence and uniqueness for complex linear Ordinary Differential Equations (ODEs) with constant coefficients, using only a result on derivatives. Unlike traditional approaches, which often rely on integration techniques, power series or the classical theorem of existence and uniqueness, we show an alternative method to obtain the general solution of these ODEs. In addition to the theoretical foundation, we explore concrete applications of these results, highlighting their relevance in physical and mathematical problems. The dissertation is organized into three main axes: the algebraic and analytical structure of complex numbers, the theory of holomorphic functions and the explicit resolution of linear ODEs of any order. Finally, we illustrate the usefulness of these concepts in applied models, such as electrical circuits and harmonic motion, demonstrating the connection between mathematical theory and practical applications.