Roughly, the equation says that the derivative of v involves the original function. This is an example of a first order linear differential equation , and I don't intend to give away the solution method right here. But I will tell you the solution, and you can check it by plugging it into the original equation. Compare the solutions in the two examples. This is approximately right, but only approximately.
Objects do gain speed as they fall.
The second solution involves an exponential. This is what we call terminal velocity .