$$(1 - x^2)y'' - 2xy' + l(l + 1)y = 0, -1 \leq x \leq 1$$
I want to go ahead and use the Sturm-Liouville theorem to prove that this equation's eigenstates are orthogonal, but it's not a given that $y'$ is bounded at $a$ or $b$, only that the D.E. is defined over $[-1,1]$. Is that sufficient?
It turns out that imposing boundedness on the solutions is equivalent to requiring that certain continuous endpoint functionals vanish on these functions. So the formalism works, and gives you discrete eigenvalues where there are eigenfunctions that are bounded near both endpoints of the interval. And the operator is self-adjoint on this domain, which gives you completeness and orthogonality of the eigenfunctions. That said, there are other conditions you can impose at the endpoints instead, and the eigenfunctions will not be bounded, but will be orthogonal and complete. The eigenvalues will be different than for the Legendre polynomials.
This problem and solution that I posed may be helpful to you: Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$
