= Solution
For each $0<\xi<2\pi$, the <Green's function> is a <distribution> in $x$ satisfying
$$
\boxed{(\partial_x^2+2\partial_x+2)g(x,\xi)=\delta(x-\xi),\qquad g(0,\xi)=0,\quad g_x(2\pi,\xi)=0.}
$$
Equivalently, $g$ is a classical homogeneous solution on each side of $\xi$, satisfies the same endpoint conditions, and obeys the <Green function jump conditions>
$$
\boxed{g(\xi^+,\xi)=g(\xi^-,\xi),\qquad g_x(\xi^+,\xi)-g_x(\xi^-,\xi)=1.}
$$
To see equivalence, let $[g]$ and $[g_x]$ denote its jumps. The singular part of $L_xg$ is $[g]\delta'(x-\xi)+([g_x]+2[g])\delta(x-\xi)$. Matching a unit <Dirac delta distribution> with no <derivative> first forces <continuity>, then the unit <derivative> jump. Conversely these conditions produce exactly that <distribution>. The solution for a regular forcing is $y(x)=\int_0^{2\pi}g(x,\xi)f(\xi)\,d\xi$. The homogeneous endpoint problem has only the zero solution, which ensures uniqueness of this <kernel>.
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