I have that:
$$ f(x)=e^{\Psi'(x)} $$
So I took the natural log of both sides:
$$ \ln(f(x))=\Psi'(x) $$
Then I integrated both sides:
$$\int \ln(f(x))dx =\Psi(x).$$
Here $f(x)$ is required to be the equation of the brachistochrone with endpoints at $(0,1)$ and $(1,1).$ (i.e. dropping a ball at the point $(0,1)$ and letting it roll across the curve to $(0,0)$ should yield the ball reaching $(0,0)$ the fastest out of any curve that has endpoints at $(0,1)$ and $(0,0)).$
I need to find $\Psi(x).$
I got stuck because I don't know how to convert the brachistochrone curve, such that it has endpoints at $(0,1)$ and $(0,0).$ Also I'm not sure if my setup of the problem is completely correct.
Edit:
There is no explicit function for $f(x).$ That being said, what explicit choice of $\Psi(x)$ will yield the curve which takes a ball from $(0,1)$ to $(0,0)$ the fastest.