 Sturm-Liouville formulation with Bessel order as eigenvalue

For fixed $\beta$, move the order term of the Bessel equation to the right: $x^2{y}^{\prime \prime }+x{y}^{\prime }+{\beta }^2{x}^{2}y=n^{2}y$. The integrating factor for its leading differential expression is $1/x$, giving the displayed Sturm-Liouville theory form with coefficient $p=x$, potential $q={\beta }^{2}x$, weight $w=1/x$ and eigenvalue $n^2$. The origin is a singular endpoint. The integrating factor must be normalized from a positive reference point; its defining logarithmic integral cannot literally start at zero. Boundary conditions are needed before this differential identity becomes a specified self-adjoint spectral problem.