= Solution
In the <distribution> definition, the <Green's function> is characterized, for fixed $0<\xi<1$, by
$$
\boxed{L_xg(x,\xi)=\delta(x-\xi),\quad g_x(0,\xi)+g(0,\xi)=0,\quad g(1,\xi)=0.}
$$
In the classical definition, solve $L_xg=0$ separately for $0<x<\xi$ and $\xi<x<1$, with those endpoint conditions, and impose
$$
\boxed{g(\xi^+,\xi)=g(\xi^-,\xi),\qquad g_x(\xi^+,\xi)-g_x(\xi^-,\xi)=4.}
$$
A jump in $g$ would introduce a <distributional derivative> of the <Dirac delta>, absent from the right side. Continuity eliminates it; integrating the equation across $\xi$ leaves one quarter of the derivative jump equal to one. These <Green function jump conditions> make the definitions equivalent. The forced solution is $y(x)=\int_0^1g(x,\xi)f(\xi)\,d\xi$.
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