Green function for Dirichlet-problem for $L=p(x)\frac{d^2}{dx^2}+q(x)\frac{d}{dx}+r(x)$

Question

Let $G(x,\xi)$ be de Green function for the Dirichlet problem for $L=p(x)\frac{d^2}{dx^2}+q(x)\frac{d}{dx}+r(x)$, where $p\in C^2$, $q \in C^1$, $r \in C^0$ and $p(x)\neq 0$ for every $x \in [a,b]$. Prove that for every $x \in [a,b]$ the function $y:\xi \mapsto G(x,\xi)$ is a solution of $L^*y=0$ on $[a,x[$ and on $]x,b]$.

Answer 1

$G^*(x,\xi)=G(\xi,x)$, so $L^*G^*(x,\xi)=L^*G(\xi,x)$. From the properties of Green's function we know that $L^*G^*(x,\xi)=0$ whenever $x$ is an element of $[a,b]$ not equal to $\xi$. Combining these two statements shows that $L^*G(\xi,x)=0$ whenever $\xi$ is an element of $[a,b]$ not equal to $x$, because the $x$ and the $\xi$ have shifted when shifting from $G^*$ to $G$. I hope this helps!