How Do You Solve the Brachistochrone Problem Using Calculus of Variations?

[jofree87]

y[1+(y')^2] = k

First solve for dx in terms of y and dy, an then use the substitution y = ksin²(θ) to obtain a parametric form of the solution. The curve turns out to be a cycloid.

My attempt:

(y')^2 = k/y-1

dy/dx = sqrt(k/y-1)

dx = dy/[sqrt(k/y-1)]

then substitute y = ksin^2(θ)

dx = dy/[sqrt(1/sin^2(θ)-1]

dx = dy/[sqrt(cos^2(θ)/sin^2(θ)]

dx = dy/cot(θ)

I don't know where to go from here, but the parametric form of y should equal k(1-cos(θ)).
Any help would be appreciated.

 

[lanedance]

shouldn't you sub in for dy in terms of dθ