Equações diferenciais ordinárias reais lineares com coeficientes constantes: uma abordagem elementar via derivadas

Abstract

This dissertation proposes an alternative didactic approach for solving Real Linear Ordinary Differential Equations with Constant Coefficients. The study aims to demystify the inherent complexity of these equations, making them accessible to students with a more elementary knowledge of calculus. The developed methodology is exclusively founded on the property that a differentiable function on an open interval is constant if, and only if, its derivative is zero. This method was detailed and applied to ODEs of first, second, and third orders, covering cases with distinct real roots, multiple real roots, and the treatment of complex conjugate roots without the explicit use of complex numbers in the derivation process. Furthermore, a brief generalization of the method for ODEs of order 𝑛 was addressed. The efficacy and versatility of the approach were corroborated through applications in physics modeling problems, such as mass-spring systems, and electrical circuits. As a pedagogical contribution, a didactic sequence was developed for high school students in mathematics olympiad classes, consolidating a practical resource for the implementation of this methodology in the classroom and fostering mathematics education more broadly.