Consider the following second-order linear and homogeneous ODE:
$$f''(x)+ \frac{\lambda}{x} \cdot f'(x) - \mu \cdot f(x) \enspace = \enspace 0$$
where $\lambda, \mu \in \mathbb{R}$. I am looking for solutions to this ODE. Unfortunately, I am not able to find them myself. I have already tried several ansatzes, but none of them succeeded. Any ideas or hints on how to find a solution to this equation?
$$ f^{\prime\prime}+\frac{\lambda}{x}f^{\prime}-\mu f=0 $$
Multiplying by $x^{2}$ gives
\begin{equation} x^{2}f^{\prime\prime}+\lambda xf^{\prime}+\left( -\mu x^{2}\right) f=0\tag{1}% \end{equation}
Bessel ode is
\begin{equation} x^{2}f^{\prime\prime}+xf^{\prime}+\left( x^{2}-n^{2}\right) f=0\tag{A}% \end{equation}
The generalized form of Bessel ode is given by (Bowman 1958) as \begin{equation} x^{2}f^{\prime\prime}+\left( 1-2\alpha\right) xf^{\prime}+\left( \beta ^{2}\gamma^{2}x^{2\gamma}-\left( n^{2}\gamma^{2}-\alpha^{2}\right) \right) f=0\tag{C}% \end{equation}
Comparing (1) and (C) shows that \begin{align} \left( 1-2\alpha\right) & =\lambda\tag{2}\\ \beta^{2}\gamma^{2}x^{2\gamma} & =-\mu x^{2}\tag{3}\\ \left( n^{2}\gamma^{2}-\alpha^{2}\right) & =0\tag{4} \end{align}
(2) gives $\alpha=\frac{1}{2}-\frac{1}{2}\lambda$. And (3) gives $2\gamma=2$ or $\gamma=1$. And (3) also shows that $\beta^{2}\gamma^{2}=-\mu$ or $\beta =\sqrt{-\mu}$. Now (4) gives $\left( n^{2}-\left( \frac{1}{2}-\frac {1}{2}\lambda\right) ^{2}\right) =0$ or $n=\left( \frac{1}{2}-\frac{1} {2}\lambda\right) $. (taking the positive root). But the solution to (C) is given by
$$ y\left( x\right) =x^{\alpha}\left( c_{1}J_{n}\left( \beta x^{\gamma }\right) +c_{2}Y_{n}\left( \beta x^{\gamma}\right) \right) $$
Therefore the solution to (1) becomes
$$ y\left( x\right) =x^{\left( \frac{1}{2}-\frac{1}{2}\lambda\right) }\left( c_{1}J_{\left( \frac{1}{2}-\frac{1}{2}\lambda\right) }\left( \sqrt{-\mu }x\right) +c_{2}Y_{\left( \frac{1}{2}-\frac{1}{2}\lambda\right) }\left( \sqrt{-\mu}x\right) \right) $$
Where $J$ is the Bessel function of first kind and $Y$ is the Bessel function of the second kind.
