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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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up vote 2 down vote accepted

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 $C_1=C_2=0$.

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OK, thanks.. :) – Jose Garcia Jul 13 '12 at 8:49

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