Please solve this differential equation problem in details $.$The Satellite Transportable Terminal $($ST $1)$ is a combat$-$proven, transportable earth terminal designed to establish secure voice, video and data communications virtually anywhere. Designed to withstand challenging conditions and demanding operations in Iraq and Afghanistan.The sprung mass of a vehicle is the mass of the vehicle minus the suspension and wheels $($the unsprung mass$).$To simplify the analysis of a suspension system, we assume that the mass of a vehicle, and any forcing function on the suspension, are evenly distributed; in other words, the suspension system can be treated as one shock.Thus, this simplified suspension system behaves according rding to the differential equation md$^2/$dt$^2+$ B dx$/$dt $+$ kx $=$ f $($t$)$ where xt$)$ is the displacement from its rest position in meters, m is the mass of the object in kg$,$ B is the damping constant in N s$/$m$,$ k is the spring constant in N$/$m$,$ and f $($t$)$ is the external force in Newtons $($N$).$The suspension system for the STTs needs to be fitted with cost effective shocks that are reliable and protect the satellite while being loaded and transported. There are three shocks available:Specifications Spring Constant k$($N$/$m$)$ Damping Coefficient B $($N s$/$m$)1$: Air Coaster $220$N$/$m $100$N$.$s$/$m$2$: cloud riders $59$N$/$m $1050$N$.$s$/$m$3$: Extreme shocks $170$N$/$m $1110$N$.$s$/$mThe trailer of a STT at rest will be loaded with a satellite with a mass of $1800$ kg$.$ Assuming that the loaded trailer is at equilibrium, the displacement of the shocks before the trailer is loaded is $3$ cm$.$It is important to determine the proper shocks for the suspension system of STTs to avoid any damage to the satellite. Topographers and engineers have determined that the force on the shocks in the most severe terrainat the highest speed can be modeled by the function f $($t$)=200$ sin$(4$t$)$ N where t is in seconds. Assume that x$(0)=0$ and x$(0)=0.$a$)$ For each shock, solve the associated non$-$homogeneous equation and plot the solutions.Find the maximum$/$minimum displacement, Xmax and Xmin for each suspension system. Discuss the shock performance in the long$-$run.Determine the best shock for the suspension system of the STT and explain your decision from the perspective of the satellite.For your recommended shock, solve the associated homogeneous equation. Explain the effect of the homogeneous solution on the satellite.