Use of complex notation for solving vibrations equations.

The article focuses on clarifying the proper use of complex numbers in solving differential equations related to vibration problems in mechanical systems. It highlights a common inconsistency in the literature where one side of the differential equation is expressed in real form while the other is complex, often without explicitly stating that the solution involves complex variables or the need to take the real part for physical interpretation. To address this, the article presents a didactic methodology using auxiliary complex variables and phasors, demonstrating that the real part of the complex solution corresponds to the physical displacement, thereby ensuring mathematical consistency and clearer understanding. Examples from discrete and continuous vibration systems illustrate this approach, emphasizing that selecting the real or imaginary part aligns the solution phase with the excitation type.