= Solution
Choose the positive causal sign convention $\mathcal L=\partial_y-x\partial_x^2$, so the forced equation is $\mathcal Lu=-xy^{-2}e^{-x/y}$. With source $(\xi,\tau)$, a forward <Green's function> satisfies
$$
\boxed{(\partial_y-x\partial_x^2)G(x,y;\xi,\tau)=\delta(x-\xi)\delta(y-\tau),\qquad G=0\text{ for }y<\tau.}
$$
It vanishes at $x=0$, decays at spatial infinity in the chosen bounded evolution class, and has the temporal jump $G(x,\tau^+;\xi,\tau)=\delta(x-\xi)$. Away from the source it solves the homogeneous equation. Equivalently one may reverse both the Green-function operator and its source sign to use the original printed operator.
To select the physical zero-initial-data solution one also excludes extra corner-source terms of the kind found in (b). With this causal bounded convention the solution representation is
$$
u(x,y)=-\int_0^y\int_0^\infty G(x,y;\xi,\tau)\frac\xi{\tau^2}e^{-\xi/\tau}\,d\xi\,d\tau.
$$
A mere pointwise statement of the three boundary limits does not by itself provide that uniqueness convention.
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