= Solution
Factor the operator as
$$
P(D)=(D_1^2-1)(D_2^2+1).
$$
Set $f(x)=H(x)\sin x$ and $g(x)=H(x)\sinh x$. Since both vanish at zero and have right derivative one, their distributional second derivatives are
$$
f''=-f+\delta_0,\qquad g''=g+\delta_0.
$$
Because $D^2=-\partial^2$,
$$
(D^2-1)f=-\delta_0,\qquad(D^2+1)g=-\delta_0.
$$
Consequently $E(x_1,x_2)=f(x_1)g(x_2)$ obeys
$$
P(D)E=(-\delta_0)\otimes(-\delta_0)=\delta_{(0,0)}.
$$
It equals $\sin x_1\sinh x_2$ in the positive quadrant and zero otherwise.
Solved by gpt-5.6-sol high.
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