= Solution
First move the order parameter term to the right-hand side:
$$
x^2y''+xy'+\beta^2x^2y=n^2y.
$$
Here $p=x^2$, $r=x$, so the <integrating factor> obeys
$$
\frac{w'}w=\frac{r-p'}p=-\frac1x,
\qquad w(x)=\frac{C}{x}.
$$
Choose the irrelevant positive constant $C=1$. Multiplying the equation by $1/x$ gives
$$
(xy')'+\beta^2xy=\frac{n^2}{x}y.
$$
Thus a <Sturm-Liouville theory> <eigenvalue> formulation with the requested <eigenvalue> is
$$
\boxed{Ly=\lambda wy,\qquad
L=\frac d{dx}\left(x\frac d{dx}\right)+\beta^2x,\quad
w=\frac1x,\quad\lambda=n^2.}
$$
This is the <Sturm-Liouville formulation with Bessel order as eigenvalue>. The spectral parameter is the squared order of the <Bessel function>, with $\beta$ treated as fixed. This differs from the also familiar formulation in which $\beta^2$ is the <eigenvalue>.
The factor $1/x$ is singular at the origin. Consequently the <integral> defining $w$ cannot literally use lower limit zero; take a positive reference point $x_*>0$, obtain $w=x_*/x$, and absorb $x_*$ into the normalization. The differential identity holds for $x>0$. <Boundary conditions> and admissibility at zero and at the outer endpoint are additional ingredients needed to make a particular self-adjoint spectral problem.
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