The solutions of for the Brachistochrone problem make use of trigonometric substitution to generate parametric equations for x and y. Does a solution exist without trigonometric substitution?
2026-04-01 01:04:12.1775005452
Is it possible to find a graphable function solution for the Brachistochrone problem without using trigonometric substitution?
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The differential equation of the cycloid profile (curvature radii ratio) when
$$ \tan \phi = y^{'} $$
is
$$ \frac{y y ^{''}}{1+y^{'2}}= -\frac12 $$
Its integral
$$ \cos \phi = \pm \sqrt{y/c} $$
involves cos function which is trigonometric... so at property level itself embedded and trig is unavoidable.