Difficult version of boatman problem

[Estudiante Curioso]

1. Problem
A boatman crosses a river of width $D$ from a point $O$, looking to get to point $A$ on the opposite riverbank. Suppose that the flowrate is uniform with velocity of magnitude $v_0$. The boat has a velocity $\vec{v_1}$ relative to the water, with constant magnitude, and it always points towards the point $A$. Calculate the time that the boatman takes to ge t to point $A$.

We were also hinted that it may be useful to state that $\vec{r_A}-\vec{r_B}=\alpha(t)\vec{v_1}$ (because $\vec{v_1}$ points to $A$), and then derive that expression.

2.The attempt at a solution
First I noticed that since $||\vec{v_1}||$ is constant, its derivative must be perpendicular to it. So the plan was to project $\vec{v_b}$ (obtained from the previous derivative) over the unitary vectors $\hat{\vec{v_1}}$ and $\hat{\dot{\vec{v_1}}}$ and then I would get two expressions from the previous derivative, from where I could potentially obtain $\alpha(t)$ through integration or a differential equation. I also noted that those unitary vectors define polar coordinates with respect to $A$.

I arrived to the expressions $$v_1+\vec{v_0} \hat{v_1} = \alpha'(t) v_1$$ and $$v_0 \hat{\dot{\vec{v_1}}}= \alpha(t) ||\dot{\vec{v_1}}||$$
where I could use trigonometry to replace the dot products on the left with $sin(\theta)$ and $cos(\theta)$ (where $\theta$ is the angle between $\vec{v_0}$ and $\vec{v_1}$), but I am completely puzzled as what to do with $||\dot{\vec{v_1}}||$ and if this method actually works.

 

[Charles Link]

I do believe this problem has the same mathematical solution as one in which point $A$ moves with a constant velocity upward, and the water has no flow of current. This same problem was given by @micromass in problem 4 of his October 2016 challenge. https://www.physicsforums.com/threads/micromass-big-october-challenge.887447/ Note: You subtract $y=v_o t$ from the y coordinate of the solution of the challenge problem. The problem gets solved, (I solved it), around posts 61-68 of the discussion. @Chestermiller also supplied portions of the solution.$\\$ Hopefully, I'm not breaking the Physics Forums rules by supplying the solution. This is really a non-trivial problem, and in general, neither the student or the homework helper would be expected to solve this problem routinely.

 

[Estudiante Curioso]

Thank you for your reply. I'm looking at the solution of the problem you posted, but I'm having a hard time understanding it. I'll keep trying.
Edit: I hadn't seen posts after 60, now it's much more clear.

 

[Charles Link]

@Estudiante Curioso See also post 89 by @Chestermiller where he completes an alternative solution to the problem.