Consider the Sturm-Liouville problem
$$y''(x) + \lambda x^2 y=0, \ y(0)=0,\ y(1)=0$$
The analytical solution is given by
$$\lambda_n=4\alpha_n^2, \ y_n(x)=\sqrt{x}J_\frac14(\alpha_nx^2)$$
where $\alpha_n$is the $n$-th zero of the Bessel function of the first kind of order 1/4. Thus, $\lambda_1\approx30.93$ and $y_1(x) \approx \sqrt{x} \,J_\frac14(2.78x^2)$
{vals, funs} =
NDEigensystem[
{-(y''[x]/x^2),
DirichletCondition[y[x] == 0, x == 0],
DirichletCondition[y[x] == 0, x == 1]},
y[x], {x, 0, 1}, 1]
yields vals = 16.1035
which is clearly incorrect.