Brachistochrone Problem

Question
If a particle falls from rest under the influence of gravity from higher to lower point in the minimum time, what is the curve that the particle will follow?
Solution
Suppose v is the speed of the particle along the curve, then in traversing ds portion of the curve time spent would be so that total time taken by the particle in moving from highest point 1 to lowest point 2 will be

Suppose vertical distance of fall upto point 2 be x, then from the principle of conservation of energy of the particle we find that

Where,

For to be minimum equation

must be satisfied. From expression for we find that

Where c’ is the constant of integration.
Since c’ is a constant we can also write

where c is also a constant. On integrating above equation we find

Putting , and on integration we get

where c” is new constant of integration.
In case c” is zero then y will be zero for x=0.
As such the equation

And this y represents an inverted cycloid with its base along y axis and cusp at the origin and is the curve that particle will follow.