One of the famous problems in the history of mathematics is the brachistochrone problem: to find the curve along which a particle will slide without friction in the minimum time from one given point P to another Q, the second point being lower than the first, but not directly beneath it. You can think of this as the shape of the track that will take a rollercoaster from point P above to point Q below.
In solving the problem, it is convenient to take the origin as the upper point P and to orient the axis so positive y is downwards and positive x is rightwards. Suppose that the lower point has coordinates (x0, y0). It is then possible to show that the curve of minimum time is given by a function y = ?(x) that satisfies the differential equation (1 + y02 )y = k2 where k2 is a certain positive constant to be determined later.
(a) Solve the above equation for y0 . Why do we have to choose the positive square root?
(b) Introduce the new variable t by the relation y = k2 sin2 t. Show that the equation found in (a) then takes the form 2k 2 sin2 t dt = dx.
(c) Letting θ = 2t, show that the solution of the equation for (b) for which x = 0 when y = 0 is given by x = k2 (θ - sin θ)/2, y = k 2 (1 - cos θ)/2.
(d) The equations in (c) are parametric equations of the solution of the differential equation presented in the problem that passes through the origin. The graph of these parametric equations is called a cycloid (maybe you've heard of these before, its what you get when you roll a small circle around a larger circle and trace the point on the outside of the small circle.) If we make a proper choice of the constant k, then the cycloid also passes through the point (x0, y0) and is the solution of the brachistochrone problem. Find k if (x0, y0) = (2, 1).
(e) What happens if the second point is directly below the starting point?