For this problem y is the variable for the displacement, v is the velocity, m is the mass, k is the constant of proportionality.
The force on the object is the mass times the acceleration: $F =$ in terms of y or in terms of the velocity we have $F =$ .
For an object in free fall there are two forces: the force due to gravity (positive g) and the force due to air resistance, which is proportional to the velocity and acts in the opposite direction. Therefore $F =$ .
Putting these two together we get a differential equation for the force acting on an object in free fall in terms of the velocity v:
$= 0$.
This equation is separable: $v^\prime = 1$, so we can integrate both sides with respect to t.
If we integrate both sides, combine all constants into one constant v0 then we get $v =$ . Integrate again we find that $y =$ , where we use y0 as the integration constant.