in section 4.2 of the book "Special functions, a graduate text" says the following:
We return to the three cases corresponding to the classical polynomials, with interval $I$, weight $w$, and eigenvalue equation of the form $$p(x)\psi_n (x) + q(x)\psi_n(x) + \lambda_n\psi_n(x) = 0,\qquad q = \frac{(pw)}{w}$$ The cases are $$I = \mathbb{R},\ w(x) = e^{-x^2},\ p(x) = 1,\ q(x) = -2x;$$ $$I = \mathbb{R}_+,\ w(x) = x^\alpha e^{-x},\ \alpha> -1,\ p(x) = x,\ q(x) = α + 1 - x$$
Also, the chebyshev polynomials are solution for the next "Sturm-Liouville" problem $$\sqrt{1-x^2}\phi''-x\phi'+n^2\phi=0,$$
Why in the above cases are no boundary conditions imposed on the differential equations?
Eigenproblems are always homogeneous, so you get homogeneous boundary conditions for the values or limit of them at the interval ends. As always, solutions have to be non-trivial, non-zero.