Analysis of solutions of differential equations of vibratory systems with varied external forces

Several Engineering problems can be modeled from differential equations, analytical and numerical methods can be employed to determine the solutions. Among these problems, applied in the area of Mechanics, there are those that involve the analysis of vibratory systems. The present work aims to conduct a study on the solutions of the second order ordinary differential equations that model these vibratory systems, seeking to solve these equations analytically from the application of different external forces. In order to solve analytically each of the equations that describe these systems, the homogeneous equation solution is first determined. Then, depending on the type of external force that acts on the system, the particular solution is obtained using the methods of Indeterminate Coefficients or Parameter Variation. The general solution is then obtained from the linear combination of homogeneous and the particular solutions. The analysis of the solutions shows that the displacements of the masses according to time, depending on the external force applied in the system, present varied behaviors among themselves. Over time, the homogeneous solution, characterized as transient response, becomes negligible, remaining only the particular solution, characterized as the permanent response.