How can the Brachistochrone problem be solved using parametric equations?

[stunner5000pt]

find the curve for which the body will follow such that the time of travel is a minimim.
Hints Minimize $$t_{12} = \int_{x_{1}}^{x_{2}} dt = \int_{x_{1}}^{x_{2}} \frac{ds}{v} = \int_{x_{1}}^{x_{2}} \sqrt{\frac{1+y'^2}{2gy}} dx$$
since F does not depend on x i can use hte beltrami identity (from the previous post)
$$H = \frac{-1}{\sqrt{2gy} \sqrt{1+y'^2}}$$
and
$$\frac{dy}{dx} = \frac{1}{2gyH^2} -1$$
this is where i am stuck
SOlving this creates an ugly mess! How can i get the parametric equations from this?

 

[dextercioby]

There's no hint, you need to set the problem right. Use conservation of total mechanical energy.

Daniel.

 

[stunner5000pt]

havent i already considered that using $$\frac{1}{2} mv^2 = mgh ==> v = \sqrt{2gh}$$??

would be very desirable to get this in terms of the parametric equations... they are far better in recognizing the cycloid