Solving Differential Equations Using Complex Variables

Аннотация

Solving differential equations is essential in many areas of science and engineering. Traditional methods, however, often become complicated when dealing with oscillatory or complex systems. This article explores how the use of complex variables simplifies the process of solving differential equations. By applying techniques such as Euler’s formula, complexification of real problems, and the Residue Theorem, complex variables provide powerful and elegant methods for finding both real and complex solutions. Several illustrative examples are presented to demonstrate the efficiency and effectiveness of these approaches. The article emphasizes the importance of mastering complex-variable methods for a deeper understanding of differential equations and their applications. Keywords: Differential equations; Complex variables; Euler’s formula; Complexification; Residue Theorem; Laplace transform; Complex analysis; Mathematical methods.