# Second order homogeneous linear ODE with variable coefficients

$\frac{d}{d x}\left(\frac{1}{1 + x^2}\frac{d f}{d x}\right) + \sigma f = 0~\Leftrightarrow~\frac{d^2 f}{d x^2} - \frac{2 x}{1 + x^2}\frac{d f}{d x} + \sigma(1 + x^2)f = 0$
for some eigenvalue $\sigma$. What I actually wish to find is an orthogonal set of solutions $f_{\sigma_i}(x)$ for some set of eigenvalues $\sigma_i$, but I would be happy only if I find solutions $f_\sigma(x)$, orthogonal set or not. What I have noticed so far is that the equation have similarities with the Legendre differential equation:
$\frac{d}{d x}\left((1 - x^2)\frac{d P}{d x}\right) + \left(\lambda(\lambda + 1) - \frac{\mu^2}{1 - x^2}\right)P = 0$
where $P_\lambda^\mu(x)$ are the associated Legendre functions, and specifically the Legendre polynomials $P_n^0(x)$ for integer $n \geq 0$ form an orthogonal set in $|x| \leq 1$. My idea was that there might be some coordinate transformation that can transform my differential equation to the Legendre form. So far I have been unsuccessful with this, and I have not yet managed to conclude whether there exists such a transformation or not. Am I on the right track? Do you know if it is possible to find my sought coordinate transformation to make it a Legendre DE, or should I rather compare with another more general DE, such as Euler's hypergeometric DE? Do you recognize my DE, and are solutions to the equation known and documented? Just about any help on this matter would be great. I have spent too many hours on this already.