= Nonunique Green kernel at a singular endpoint with two vanishing modes
{title2=$Ly=y''-3y'/x+3y/x^2$}
On $0<x<1$, both independent homogeneous solutions $x,x^3$ vanish at the singular endpoint zero. Consequently imposing $y(0)=y(1)=0$ leaves the nonzero homogeneous solution $x^3-x$ and does not determine a unique inverse. One valid right <Green function>, with $L_xG=\delta(x-z)$ and zero endpoint values, is
$$
G_0(x,z)=\begin{cases}x(z^3-z)/(2z^3),&x\leq z,\\(x^3-x)/(2z^2),&x\geq z.\end{cases}
$$
Its first <derivative> has jump one because the <Wronskian> of $x$ and $x^3-x$ is $2z^3$. Every kernel $G_0+(x^3-x)h(z)$ preserves these properties when the integrals exist. For sources supported away from zero, integration yields one particular solution; the full endpoint-constrained family adds $C(x^3-x)$. An additional endpoint selection, such as $y(x)/x\to0$, can choose a unique member. This is a singular-endpoint phenomenon, not an ordinary regular two-point uniqueness theorem.
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