With , we have by the fundamental theorem of calculus
Differentiating both sides wrt gives
This equation is separable as
Integrate both sides; on the left, substitute
Given that , we have
So the velocity at time is that satisfies
When , we have
We can rewrite the particular solution as
Taking the limit as on both sides gives
(the exponential terms approach 0)
so the limiting velocity, call it , satisfies the quadratic equation
Realistically, the boat won't speed up enough for the resistance to be so strong as to reverse the boat's direction, so the limiting velocity should be positive.