Use of complex notation for solving vibrations equations

Complex numbers and complex functions are frequently utilized in solving equations of motion in free and harmonic responses of vibration systems. However, certain fundamental details are sometimes overlooked and many books and articles may present differential equations with inconsistencies, which can confuse meticulous readers. This confusion often arises because one side of the equations is in real form, while the other side is expressed in complex form. To address this issue, the present article offers examples from literature to highlight the perplexing situation and provides a didactic methodology, aimed at enhancing clarity and ensuring that readers are left with a thorough understanding of the subject matter.