How do I solve the Brachistochrone Problem with a given differential equation?

[930R93]

Hello,
I'm having problems with a D.E. question,
I'm asked to solve the equation:
$$\left(1+y^{'2}\right)y=k^{2}$$
where K is a certain positive integer to be determined later.
This more commonly known, as you probably know, as one of the solutions to the Brachistochrone Problem.
I really have no idea where to start. I've experimented with various methods of integration, which create nothing which I recognized as productive.
A gentle prod in the right direction would be much appreciated!
Thanks!
-930R93

 

[gato_]

Integrate $$y'=\sqrt{(k^{2}-y)/y}$$. It is useful to use $$y=k^{2}sin^{2}(theta)$$ .