= Constant forcing in a logarithmic Dirichlet Sturm-Liouville problem
{title2=$-x^2y''-xy'-\alpha y=1,\quad y(1)=y(e^\ell)=0$}
For $\ell>0$, the <logarithmic change of variable in a Sturm-Liouville problem> sends this equation to $-Y''-\alpha Y=1$ on $(0,\ell)$. The normalized <eigenfunctions> are $\sqrt{2/\ell}\sin(n\pi t/\ell)$, with <eigenvalues> $\lambda_n=(n\pi/\ell)^2-\alpha$. If every $\lambda_n$ is nonzero, the <Sturm-Liouville eigenfunction expansion> gives
$$
\boxed{y(x)=\sum_{\substack{n\geq1\\n\text{ odd}}}\frac{4\sin(n\pi\log x/\ell)}{n\pi[(n\pi/\ell)^2-\alpha]}.}
$$
Indeed the constant function has sine coefficients $2[1-(-1)^n]/(n\pi)$; division by each eigenvalue solves the equation in the operator domain. The series is uniformly convergent, with coefficients $O(n^{-3})$, while applying the differential operator is justified in the weighted square-integrable sense. At a zero eigenvalue the <solvability condition at a Sturm-Liouville eigenvalue> must instead be checked: an odd resonant index makes the constant forcing incompatible, whereas an even resonant index permits solutions with an arbitrary multiple of that zero mode.
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