Solution to 2nd-order homogeneous linear ODE with variable coefficients. 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?
 A: $$
f^{\prime\prime}+\frac{\lambda}{x}f^{\prime}-\mu f=0
$$
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.
