Green kernels for a radial dipole Euler equation ($L={\partial }_x^2+x^{-1}{\partial }_x-x^{-2}$)

The homogeneous Cauchy-Euler equation $Ly=0$ has fundamental solutions $x,x^{-1}$ on $x>0$, with Wronskian $-2/x$. Requiring decay at both endpoints gives the unweighted Green's functionRequiring zero initial value and derivative at zero instead gives the forward kernelBoth kernels are continuous and have first-derivative jump one at $x=\xi$, so $L_{x}G=\delta (x-\xi )$. Away from that point they solve the homogeneous equation. Integrating against the forcing with ordinary measure $d\xi$ yields the respective solutions whenever the integral and endpoint limits converge. Their difference is $G_0-G_D=x/2$, independent of $\xi$, so the solutions differ by the homogeneous term $(x/2)\int f(\xi )d\xi$ for integrable forcing. For example, the two choices for an indicator forcing on $(0,1)$ differ by $x/2$. Specifying a differential operator without its endpoint conditions does not specify its Green kernel.