= Solution
Use the <weighted inner product> $\langle f,g\rangle_w=\int_a^bw(x)\overline{f(x)}g(x)\,dx$ with positive weight $w$. For a <self-adjoint operator> $L$ and nonzero <eigenfunction> $Ly=\lambda y$, self-adjointness gives $\lambda\langle y,y\rangle_w=\langle y,Ly\rangle_w=\langle Ly,y\rangle_w=\overline\lambda\langle y,y\rangle_w$. The positive norm therefore forces $\lambda$ to be real. For two <eigenfunctions> of distinct <eigenvalues>, the same calculation gives $(\lambda_2-\lambda_1)\langle y_1,y_2\rangle_w=0$, proving their <orthogonality>.
The <Chebyshev differential equation> takes the <Sturm-Liouville form>
$$
-\frac d{dx}\left(\sqrt{1-x^2}\,y'\right)=n^2\frac{y}{\sqrt{1-x^2}},\qquad
\boxed{p=\sqrt{1-x^2},\quad w=(1-x^2)^{-1/2},\quad q=0.}
$$
Its operator is $Ly=-(1-x^2)y''+xy'$, so $Lx=x$ as required. For $y=x^3+Bx$, direct substitution gives $Ly=9x^3+(B-6)x$. Comparing coefficients yields
$$
\boxed{y_3=x^3-\frac34x,\qquad n^2=9.}
$$
Explicitly, using $x=\cos\theta$,
$$
\langle y_1,y_3\rangle_w=\int_{-1}^1\frac{x^4-3x^2/4}{\sqrt{1-x^2}}dx=\frac{3\pi}8-\frac34\frac\pi2=0.
$$
The endpoints are singular because $p$ vanishes and $w$ diverges there. The polynomial solutions use the natural condition $py'\to0$ at each end; under $x=\cos\theta$ these are the <Neumann eigenfunctions of the Chebyshev differential operator>, proportional to $\cos(n\theta)$. They differ from the zero-value endpoint realization of the same differential expression.
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