= Solution
{c}
Expand in the <Fourier sine basis>, using unnormalized <eigenfunctions>:
$$
f(x)=\sum_{n=1}^\infty f_n\sin\frac{n\pi x}{l},\qquad
f_n=\frac2l\int_0^lf(t)\sin\frac{n\pi t}{l}\,dt,
\qquad y(x)=\sum_{n=1}^\infty c_n\sin\frac{n\pi x}{l}.
$$
Projection of $\mathcal Ly=f$ onto each <eigenfunction> gives
$$
\boxed{\left[1-\left(\frac{n\pi}{l}\right)^2\right]c_n=f_n.}
$$
When $l\notin\pi\mathbb N_{>0}$, every <eigenvalue> is nonzero and the <eigenfunction expansion> is uniquely determined:
$$
\boxed{y(x)=\sum_{n=1}^\infty
\frac{f_n}{1-(n\pi/l)^2}\sin\frac{n\pi x}{l}.}
$$
When $l=m\pi$, the exceptional index is $n=m$, for which the <eigenfunction> is $\sin x$. If $f_m\ne0$, its coefficient equation is $0=f_m$, an inconsistency: there is no solution. If $f_m=0$, the coefficient equation is $0=0$, and $c_m$ is arbitrary. The full family is
$$
\boxed{y(x)=C\sin x+
\sum_{\substack{n\geq1\\n\ne m}}
\frac{f_n}{1-(n/m)^2}\sin\frac{nx}{m},\qquad C\in\mathbb R.}
$$
This is the <Fredholm alternative> directly in the coefficient equations: nonzero <eigenvalues> fix their coefficients; a zero <eigenvalue> requires zero forcing in that direction and leaves the corresponding coefficient free. An inconsistent equation at a zero <eigenvalue> cannot be repaired by assigning an infinite coefficient.
For convergence, <Lipschitz continuity> implies <absolute continuity> of $f$, with bounded <derivative> almost everywhere. One <integration by parts> gives $f_n=O(n^{-1})$; since the denominators grow quadratically, $c_n=O(n^{-3})$ apart from the single free coefficient. The solution series and its first <derivative> therefore converge uniformly. The second <derivative> converges in $L^2$, and the resulting <Sobolev space> solution satisfies $y''=f-y$. As $f-y$ is a <continuous function>, this is a classical $C^2$ solution, with both <boundary conditions> satisfied. The forcing's <Fourier sine series> is interpreted on the open interval or in $L^2$; its endpoint values need not equal the endpoint values of $f$.
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