A classical problem in the calculus of variations is to find the shape of a curve such that a bead, under the influence of gravity, will slide from point A(0,0) to point B (x1,y1) in the least time. It can be shown that a nonlinear differential for the shape y(x) of the path is $\displaystyle y[1 + (dy/dx)^2] = k$, where k is a constant. First solve for dx in terms of y and dy and then use the substitution $\displaystyle y = ksin^2(theta)$ to obtain a parametric form of the solution. The curve turns out to be a cycloid.

Not entirely sure what to do with this problem, mainly b/c I don't really understand what the problem is asking, which includes what a cycloid is...or how you find the parametric solution for one.

Not sure if this is right, but it's what I've done so far:

$\displaystyle (dy/dx)^2 = k/y - 1$

$\displaystyle dy/dx = (k/y - 1)^1^/^2$

$\displaystyle dx = dy/[(k/y - 1)^1^/^2]$

Thanks in advance.