Minimizing the Functional for the Brachistochrone Problem

[WackStr]

Homework Statement

So if +x points downward and +y points rightwards then the functional that needs to be minimized is:

$$\sqrt{2g}T[y]=\int_{x_0}^{x_1}\frac{dx}{\sqrt{x}}\sqrt{1+\left(\frac{dy}{dx}\right)^2}$$

Homework Equations

I think we just have to use the Euler lagrange equation. The book (Hand and Finch says) the solution is:

$$y(x)=\sqrt{x(2r-x)}+2r ArcSin\left(\frac{x}{2r}\right)$$

This is not even a brachistrone curve! Am I missing something?

 

[member 553083]

The Attempt at a SolutionI have tried to use the Euler Lagrange equation but I just can't seem to get the answer that is given. Any help would be greatly appreciated!