= Green kernels for a radial dipole Euler equation
{c}
{title2=$L=\partial_x^2+x^{-1}\partial_x-x^{-2}$}
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 function>
$$
G_D(x,\xi)=\begin{cases}-x/2,&x\leq\xi,\\-\xi^2/(2x),&x\geq\xi.\end{cases}
$$
Requiring zero initial value and derivative at zero instead gives the forward kernel
$$
G_0(x,\xi)=\begin{cases}0,&x\leq\xi,\\(x-\xi^2/x)/2,&x\geq\xi.\end{cases}
$$
Both kernels are continuous and have first-derivative jump one at $x=\xi$, so $L_xG=\delta(x-\xi)$. 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.
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