Frobenius & Special Functions · Challenge problem

The equation $x^2y'' - 2xy' - 4y = 0$ has indicial roots $4$ and $-1$. For the solution with $y(1)=5$, $y'(1)=0$, the two terms $c_1x^4$ and $c_2x^{-1}$ are equal at some $x>1$. Find that value of $x$. Give an exact answer.