Abstract: The problem of SDOF response under harmonic load is usually faced the engineers in their design life. SDOF represents many structures in practice such as elevated liquid tower or tank. Any equipments or machines inside structure can exert a harmonic loading on the structure itself. The objective of this research is to find the complete solution to the differential equation of SDOF structure under sinusoidal loading. Further, state when the designer or mathematician can neglect the homogonous part of the total solution and when this part has a significant effect on the total solution. In this study, mathematics of the differential equation for particular solution is extended to include the homogenous solution. MATLAB was used to plot the homogenous, particular and the total solutions for the covering differential equation. Many solution variables (excitation frequency, loading frequency ratio and viscous damping ratio) were used to test the results obtained and generalized the solution. From the data collected, it was concluded when the homogenous solution part was neglected from the total solution, the results will have an error which depends on the value of the damping ratio and the time at which the total solution evaluated. If the value of the damping ratio is small, large window of time required to minimums the error. While a small windows of time required when the damping ratio has a larger values.