@Rouben Rostamian Thank you for reading and commenting on my worksheet.
After examining your solution to the least time crossing, here are a few observations:
I animated the boat (as a plot arrow) as it crossed the river following the least time path. It always headed upstream (against the current) with remarkably little variation in its bearing (the value of alpha).
I tried reducing the boat's speed, but any value below 0.935 produced this error message from dsolve;
Error, (in dsolve/numeric/bvp) initial Newton iteration is not converging
This same error occurred if I moved the destination upstream from the starting point, however a destination downstream from the starting point produced a correct least time path.