ODE Derivation (bead on a frictionless wire) Which Maximises Speed

[Idealistic]

A bead, strung on a frictionless wire and acted upon only by gravity, is to slide down the wire between two fixed end points. Determine the shape of the wire which minimizes the time of the descent. (You only need determine the ode which the shape must satisfy). You may want to use the fact that the speed of the bead is given by v = sqrt(2gy(x)) and the arc-length travelled by the bead is given by ds = sqrt(1 + y'(x)²) If the bead starts at y = 0 and the y-axis is pointed down.

I usually begin these problems by starting with Newton's second law:

F = ma; and since gravity is the only force acting on the bead in this example,

-mg = m*dv/dt = m*dv/dy*dy/dt = -m*g/sqrt(2gy(x))*dy/dt

I'm not sure what to do next though, or how to incorperate the arc length. Do I change variables variables? Do I have to look at v and maximize it in terms of y as a function of s?​

 

[galactus]

This appears to be a problem involving the 'Brachistochrone problem'

Google it and you ill find plenty.

This is a famous problem solved by Newton.