= Solution
Only two coefficients in the <Sturm-Liouville eigenfunction expansion> are nonzero: $a_0=2$ and $a_1=2-\alpha$. Since $\alpha\ne2$, the result of part (i) simplifies to
$$
Y=\frac{2}{1-\alpha}y_0+\frac{2-\alpha}{2-\alpha}y_1=\frac{2}{1-\alpha}y_0+y_1.
$$
Using <Parseval's identity>, the prescribed <weighted norm> becomes
$$
\frac{4}{(1-\alpha)^2}+1=2.
$$
Thus $1-\alpha=\pm2$, so $\alpha=-1$ or $3$. Positivity selects
$$
\boxed{\alpha=3,\qquad Y=-y_0+y_1.}
$$
The standing nonresonance assumption still applies: this answer requires that $3$ is not one of the unspecified higher <eigenvalues>. Their actual values are not supplied. If a higher <eigenvalue> were $3$, the stated exclusion would fail and the data would not have an admissible $\alpha$ under that exclusion.
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