Use the method in this two topic: Models for Population Growth and Linear Equations A skydiver's velocity through the air can be modeled by the differential equation $\backslash ($ m $\backslash$frac$\{$d v$\}\{$d t$\}=$m g$-$k v $\backslash ),$ where $\backslash ($ k $\backslash )$ is a constant called the drag coefficient $($dependent primarily on body orientation$),\backslash ($ m $\backslash )$ is the mass of the skydiver, and $\backslash ($ g $\backslash )$ is earth's gravitational acceleration of $\backslash (9\mathrm{.}8\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash ).$

a$.$ Show that the differential equation is linear. Then find the specific solution $\backslash ($ v$($t$)\backslash ),$ using the initial condition $\backslash ($ v$(0)=0\backslash ).$

b$.$ Assume this skydiver has a mass of $75$ kg $,$ and they are positioned so their drag coefficient is $\backslash ($ k$=1\backslash ).$ Find their velocity after having fallen for $5$ seconds after leaving the plane, and how far they have fallen at that point.

c$.$ Objects accelerating under sufficient drag have what's called a terminal velocity $-$ this is the maximum speed it can reach, because eventually the drag becomes so great that it negates the acceleration. Find this skydiver's terminal velocity if they maintain their body orientation.