# stuck on a differential equation

let be the differential equation

$x^{2} y''(x)+ y(x)(a^{2}+k^{2} _{n})=0$

the boundary conditions are $\int_{0}^{\infty}dx |y(x)|^{2} < \infty$ and $y(0)$ must be finite (regular solutions near the origin )

here $a^{2} >0$ and $k^{2} _{n}>0$ these $k_{n}$ are a discrete set of eigenvalues

my question is how can i transform my differential equation into a more well-known differential equation so i can get the eigenvalues

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It is an homogeneous Euler equation. Its solution is $y(x)=C_1x^{r_1}+C_2x^{r_2}$ where $r_1,r_2$ are the solutions of the indicial equation $$r(r-1)+a^2+k_n^2=0.$$ None of these solutions is square integrable on $(0,\infty)$ unless $C_1=C_2=0$.