Please help me spot my mistake:
I have an equation (u(x)^{-2} + 4u'(x)^2)^{\frac{1}{2}} - u'(x)\frac{d}{du'}(u(x)^{-2} + 4u'(x)^2)^{\frac{1}{2}} = k where $k$ is a constant.
I am quite sure that if I take $u(x) = \sqrt{y(x)}$ I would have the brachistochrone equation, hence I am expecting a cycloid equation if I let $u(x) = \sqrt{y(x)}$ in the result, but I don't seem to get it :(
My workings are as follows: u(x)^{-2} + 4u'(x)^2- 4u'(x)^2 = k \times (u(x)^{-2} + 4u'(x)^2)^{\frac{1}{2}} \implies u(x)^{-4} = k^2 \times (u(x)^{-2} + 4u'(x)^2) \implies u'(x)= \frac{1}{2k}\sqrt{u(x)^{-4} - k^2u(x)^{-2}} $\implies \int \frac{1}{u \sqrt{u^2 - k^2}} du = \int \frac{1}{2k} dx$ Change variable: let $v = \frac{u}{k}$ $\implies \int \frac{1}{v \sqrt{v^2 - 1}} dv = \frac{x+a}{2}$, where $a$ is a constant $\implies \operatorname{arcsec}(v) = \frac{x+a}{2} $ $\implies \operatorname{arcsec}\left(\frac{\sqrt{y}}{k}\right) = \frac{x+a}{2}$ which does not seem to describe a cycloid...
Help would be very much appreciated! Thanks.