Frobenius & Special Functions · Challenge problem

For $x^2y''+xy'+9y=0$ ($x>0$), the solution with $y(1)=1$, $y'(1)=0$ is $y=\cos(3\ln x)$, which is next zero at some $x_1>1$. For $x^2y''+xy'+100y=0$ with the same initial data, the analogous solution $\cos(10\ln x)$ is next zero at some $x_2>1$. Find the exact value of $x_1/x_2$.