= Solution
For the <Green function> of $D_x^2-1$, the homogeneous solutions satisfying the left and right endpoint conditions are $u(x)=\sinh x$ and $v(x)=\sinh(1-x)$. Their <Wronskian> is
$$
W=uv'-u'v=-\sinh1.
$$
Continuity of the kernel at $x=\xi$ and <integration> of $G_{xx}-G=\delta(x-\xi)$ require $[G_x]_{\xi-}^{\xi+}=1$. Hence the <Dirichlet modified Helmholtz Green function on an interval> is
$$
\boxed{G(x,\xi)=-\frac{\sinh(x_<)\sinh(1-x_>)}{\sinh1}.}
$$
Here $x_<=\min(x,\xi)$ and $x_>=\max(x,\xi)$. The negative sign is essential for this operator convention. Integrating the forcing against the kernel gives
$$
\boxed{y(x)=-\frac1{\sinh1}\left[\sinh(1-x)\int_0^x\sinh\xi\,f(\xi)\,d\xi+\sinh x\int_x^1\sinh(1-\xi)f(\xi)\,d\xi\right].}
$$
This satisfies both endpoint conditions. It solves the equation classically for continuous $f$, and in the corresponding weak sense for integrable forcing. A homogeneous difference satisfying both endpoint conditions must vanish, so the solution is unique.
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