Working in a physic framework I came up with the following differential equation \begin{equation} \frac{\partial}{\partial x} \left( \frac{1}{x} \frac{\partial}{\partial x} f(x,y) \right) + \frac{y^2}{x} f(x,y) = 0, \quad f(\epsilon,y)=1, \quad \frac{\partial}{\partial x}f(x_0,y) = 0. \end{equation}
Here $\epsilon$ must be undertood as a limit ($\epsilon \to 0$) and $x_0 \in \mathbb{R}$ is some constant value. Performing a naive change of variables $g(x,y) = f(x,y)/x$ the equation now reads \begin{equation} \frac{\partial^2}{\partial x^2} g(x,y) + \frac{1}{x} \frac{\partial}{\partial x}g(x,y) + \left(y^2 - \frac{1}{x^2} \right) g(x,y) = 0 \end{equation}
which looks like a modified Bessel equation. However, I have never studied such differential equation type and I got stucked trying to solve it. Any help?
It's not modified: it's essentially the ordinary Bessel differential equation for $n=1$ after the change of variables $t = xy$. The general solution is $$ g(x,y) = c_1(y) J_1(xy) + c_2(y) Y_1(xy) $$ where $J_1$ and $Y_1$ are Bessel functions of the first and second kinds.