I've been trying to tackle a nonlinear first order differential equation that appears when trying to solve the brachistochrone problem through earth's gravitational field. Although being a long and tedious calculation, I managed to find the extremizing path in polar coordinates that lead to a hypocycloid curve. However, in the literature, I found the solution in cartesian coordinates which seems to be easier to solve but I cannot tackle a seemingly simple nonlinear differential equation. Namely,
$$ (dx)^2+(dy)^2=C^2(R^2-x^2-y^2)$$
Where $C$ and $R$ are constants.
I've been tryng the typical separation of variables, integrating factor but I convinced myself that it must be some change of variable that I haven't been able to find.
Anyways, if someone knew the starting point for tackling this equation, I would be able to sleep better tonight :) By the way, I don't need the solution which I already know, it is the method I should use that I am interested in.
Thank you very much!