@rlopez 

It is solve which returns infinitely many solutions. Yes, if only the principal branch of ln is to be considered (as Maple does) then there are only two solutions, because the equation is equivalent to  ln(z) = k*Pi + k*Pi*I  (k in Z), which is possible only for k=0 and k=1 (argument(z) = k*Pi must be in (-Pi,Pi]).

# X(10^7) = 29529388;
# X(10^8) = 213554980;
# X(10^9) = 92599500093871447;

(Edited)

@Earl 

You may fine-tune the dsolve parameters. E.g. adding method=bvp[midrich]
is enough to work for a smaller speed.

@Christopher2222 

The following list has a simple and clear pattern. But probably it is (almost) impossible to find (guess) the next number.

1, 4, 10, 19, 31, 41, 49, 59, 78, 109, 143, 166, 173, 178, 204, 259, 328, 378, 388, 377, 388, 457, 570, 668, 701, 672, 647, 703, 850, 1018, 1106, 1078, 1004, 1009, 1161, 1401, 1582, 1596, 1483, 1403, 1509, 1797, 2096, 2210, 2097, 1922, 1923, 2201, 2613, 2882, 2841, 2594, 2445, 2634, 3111, 3565, 3681, 3433, 3124, 3137, 3592, 4215, 4566, 4421, 3996, 3773, 4087, 4806, 5429, 5511, 5072, 4605, 4659, 5346, 6215, 6628, 6325, 5678, 5390, 5881, 6893, 7688, 7687, 7002, 6362, 6498, 7477, 8618, 9061, 8538, 7632, 7301, 8031, 9382, 10338, 10197, 9210, 8396, 8665, 9999, 11426, 11856, 11045, 9853, 9513, 10551, 12280, 13375, 13027, 11686, 10707, 11174, 12923, 14642, 14999, 13834, 12336, 12037, 13455, 15596, 16793, 16161, 14420, 13300, 14040, 16263, 18263, 18479, 16889, 15080, 14883, 16758, 19334, 20580, 19585, 17405, 16183, 17277, 20028, 22286, 22280, 20200, 18085, 18066, 20474, 23497, 24726, 23282, 20635, 19365, 20903, 24226, 26704, 26386, 23754, 21355, 21601, 24618, 28084, 29217, 27237, 24106, 22857, 24933, 28866, 31508, 30781, 27541, 24897, 25503, 29202, 33092, 34037, 31436, 27818, 26673, 29384, 33950, 36687, 35446, 31554, 28718, 29792, 34237, 38517, 39168, 35864, 31772, 30830, 34272, 39482, 42226, 40363, 35786, 32829, 34486, 39733, 44351, 44594, 40508, 35973, 35344, 39613, 45461, 48110, 45517, 40234, 37245, 39603, 45697, 50583, 50293, 45358, 40427, 40236, 45421, 51883, 54320, 50888, 44898, 41982, 45163, 52133, 57199, 56246, 50405, 45146, 45524, 51708, 58743, 60837, 56462, 49778, 47057, 51183, 59046, 64183, 62432, 55641, 50141, 51232, 58486, 66031, 67640, 62225, 54881, 52492, 57683, 66433, 71518, 68832, 61063, 55428, 57380, 65764, 73735, 74705, 68162, 60214, 58308, 64677, 74291, 79182, 75425, 66669, 61026, 63990, 73547, 81840, 82011, 74263, 65788, 64528, 72182, 82615, 87152, 82194, 72461, 66956, 71084, 81838, 90328, 89533, 80520, 71618, 71178, 80211, 91393, 95405, 89120, 78445, 73240, 78683, 90639, 99178, 97248, 86927, 77721, 78281, 88773

@Rouben Rostamian  

Yes, I made a mistake, unfortunately I did not check the statement about uniquness. y(1-x) is a path joining the points (0,0) and  (1,0) but not an admissible one (the boat will not have a constant speed here).
Also, because -Pi/2 < alpha, psi < Pi/2, it is not difficult to see that psi-alpha = Pi - arcsin(k)   cannot occur.
Sorry, my excuse is that I'm on holyday :-)

The optimal path is not unique in general: if y(x) is an optimal path, then y(1-x) is optimal too because the boat can go back on the same path in the same amount of time.

Furthermore, when V(x) is symmetric wrt x=1/2 i.e. V(x)=V(1-x) (as in the provided parabolic example) then it seems that  y(x) = - y(1-x).

[In the worksheet, from sin(psi-alpha)=k ==> psi-alpha = arcsin(k) OR  psi-alpha = Pi - arcsin(k) ...].

[Edited]

@Rouben Rostamian  

Excellent presentation. Congratulations!

A bare definition appears here.

I think that a good idea would be to have an easy access to a list (database) of known bugs + workarounds (not necessarily patches) if available.

The definition is contained in the worksheet (formula before (3))
such that anyone can see whether it suits his/her needs (and adapt it if necessary).
Otherwise it could have been given directly as:

L:=z->(sqrt(2)*exp(-z^2/2)+z*sqrt(Pi)*(erf(z*sqrt(2)/2)-1))/(2*sqrt(Pi));

On the other side, it would be nice a higher level of politeness in comments.

@tomleslie 

Probably (only?) acer can do it :-)

Please try to formulate mathematically the problem (forget about Maple for the moment).
For example:

Find the C^1 functions f(t,x,p) defined on R^3 such that

   f(t,x(t),x'(t)) = x'(t) * (∂f/∂p)(t,x(t),x'(t))

for any C^1 function x(t) defined on R.

For this problem the solution would be

f(t,x,p) = p * C(t,x),  where C is an arbitrary C^1 function on R^2.

Now, try to formulate in this manner your problem.

@asa12 

You must define/explain it. Use quantifiers.

@mikemeson 

In the above example the Array was obtained of course by a simple procedure, but you may define it as complicated as you want and then use surfdata. I don't understand your objection.

@tomleslie 

I posted once a modification like yours including the modified proc() and the post was deleded because of "Copyright" problems.
I did not understand why it was considered so, but this was the fact. A patch like mine (even almost  identical to yours) seems to be acceptable.