 Nonunique Green kernel at a singular endpoint with two vanishing modes

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 $L_{x}G=\delta (x-z)$ and zero endpoint values, isIts 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 $y(x)/x\to 0$, can choose a unique member. This is a singular-endpoint phenomenon, not an ordinary regular two-point uniqueness theorem.