An object weighing $4$ kg$,$ one end of which is connected to the other end of a spring connected to the ceiling, makes an oscillating movement in a frictional environment. It is known that the friction force is proportional to the speed of the object and its proportionality coefficient is $5$ Ns$/$m and the spring constant is $0\mathrm{.}2$ N$/$m$.$ At t $=0,$ the object starts to move after being raised $2$ m upwards from the equilibrium position, and it is known that it passes $5$ m below the equilibrium position at t $=10$ s$.$ Write the differential equation that describes the motion of the object and then find the equation that relates the position of the object to time.