Frobenius & Special Functions · Challenge problem

A Cauchy–Euler equation $x^2y'' + pxy' + qy = 0$ has the particular solutions $y=x^2$ and $y=x^{-3}$ (so its indicial roots are $2$ and $-3$). For the solution with $y(1) = 5$ and $y'(1) = 1$, find $y(2)$. Express your answer as a common fraction.