# The uniqueness of the brachistochrone

How does one show the uniqueness of the solution to the brachistochrone problem? Doesn't the fact that the solution is of the form $x=a-c(2t+\sin2t)$ and $y=c(1+\cos2t)$ naturally guarantee uniqueness given the 2 endpoints of the path -- 2 unknowns $(a,c)$ and 2 restraints (the 2 endpoints)?

Thanks!

-
Sure, you have those conditions. Now, how sure are you an algebraic curve doesn't suit the bill? That's where solving the associated differential equations comes in... –  Guess who it is. Jul 27 '11 at 9:26
Hi, J.M., I'm afraid I don't quite understand... Would you mind elaborating a bit? Thanks! –  Robin H Jul 27 '11 at 9:37
I can for instance construct a parametric cubic which matches the boundary conditions, but does not satisfy the brachistochrone condition... –  Guess who it is. Jul 27 '11 at 9:41