I'm trying to find the soltution of the brachistochrone problem. I'm was wondering if there is a way to solve it directly using Mathematica. The problem is the following:

[![enter image description here][1]][1]

I have to find the equation y(x) that minimizes the functional J[t]. This functional(y(x)) is the path of minimum time that a body will perform from a point y1 to a point y2.

To achieve that, it is needed to use the Euler equation:

[![enter image description here][2]][2]

where

[![enter image description here][3]][3]

The code below is the solution of the Euler equation that i was trying to integrate without sucess.

    Integrate[(-1 - y'[x]^2 - 2 y[x] y''[x])/(
     2 y[x]^(3/2) (1 + y'[x]^2)^(3/2)), {y[x], 0, 1}]

This code returns:

     Integral of (-1-(y^\[Prime])[x]^2-2 Integrate`$$a$201005 (y^\[Prime]\[Prime])[x])/Integrate`$$a$201005^(3/2) does not converge on {0,1}. >>

Thanks in advance!


  [1]: https://i.sstatic.net/fhbwp.png
  [2]: https://i.sstatic.net/NZcLU.png
  [3]: https://i.sstatic.net/hDE47.png