= Solution
To obtain <Sturm-Liouville form>, multiply the equation by $(1-x^2)^{-1/2}$. The derivative identity
$$
\frac d{dx}\big[\sqrt{1-x^2}\,y_n'\big]=\sqrt{1-x^2}\,y_n''-\frac{x}{\sqrt{1-x^2}}y_n'
$$
gives
$$
\boxed{-\frac d{dx}\left[\sqrt{1-x^2}\,y_n'\right]=n^2\frac{y_n}{\sqrt{1-x^2}},\qquad p(x)=\sqrt{1-x^2},\quad q(x)=0,\quad w(x)=\frac1{\sqrt{1-x^2}}.}
$$
The weight determines the <weighted inner product> on $(-1,1)$. It is integrable but singular at the endpoints, where $p$ also vanishes. Thus this is a singular-endpoint <Sturm-Liouville problem>, not a regular problem with strictly positive endpoint values of $p$. The transformation in part (ii) specifies its admissible Dirichlet realization; these are the <Dirichlet eigenfunctions of the Chebyshev differential operator>.
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