Formulate the differential equation for a free$-$falling body of mass m in air by taking air resistance into consideration. Ignore the effect of buoyancy and suppose that the force due to drag is given with the term $2$ with k the drag coefficient $(>0)$ and u the velocity of the free$-$falling body. Then answer the following questions: $($a$)$ Find the solution of the differential equation as a function of m$,$ k$,$ g $($acceleration of gravity$).($b$)$ Graph the solution subject to the initial condition u$($t$=0)=0.($c$)$ Does the differential equation have a critical point $($steady state solution$)?$ What does this point represent? $($d$)$ Give an estimate of the time it takes for the free$-$falling body to attain its terminal velocity. $($e$)$ If distance s is measured from the point where the mass was released above the ground, is related to velocity u by $=(),$ find an explicit $$ expression for s$($t$)$ if s$($t$=0)=0$