= Solution
Write the equation as $(L+\beta)y=-\cos x/x$ with the positive <Sturm-Liouville operator> from part (a). A <Sturm-Liouville eigenfunction expansion> therefore gives $a_n=-f_n/(4n^2+\beta)$, where weighted <orthogonal projection> gives
$$
f_n=\frac4\pi\int_0^{\pi/2}\sin(2nx)\cos x\,dx
=\frac{8n}{\pi(4n^2-1)}.
$$
Here the <product-to-sum formulas> reduce the <integral> to half the sum of the integrals of $\sin((2n+1)x)$ and $\sin((2n-1)x)$. Consequently
$$
\boxed{y(x)=-\sum_{n=1}^\infty\frac{8n}{\pi(4n^2-1)(4n^2+\beta)}\frac{\sin(2nx)}x.}
$$
At $x=0$, use the continuous limits of the terms. The coefficients are $O(n^{-3})$, so the resulting endpoint series converges. Completeness is the ordinary <Fourier sine series> completeness after $v=xy$. For an independent check, solving $v''-\beta v=\cos x$ with zero endpoint values gives
$$
y(x)=\frac{\sinh(\sqrt\beta(\pi/2-x))/\sinh(\sqrt\beta\pi/2)-\cos x}{(1+\beta)x}.
$$
Its numerator vanishes at zero, and it has the same <orthogonal projection> coefficients. Positivity of $L+\beta$ proves <uniqueness>.
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