Green kernel with a vanishing Wronskian at an interior singular point ($G(x,\xi )=u(\min (x,\xi ))v(\max (x,\xi ))/W(\xi )$)

The usual Green function formula for a monic second-order equation requires a nonzero Wronskian at the source. At an interior singular point it may fail to define an ordinary inverse for arbitrary forcing. For example, for $Ly=y^{\prime \prime }-(1+x)y^{\prime }/x+y/x$ on $[-1,1]$, choose $u=1+x$ and $v=e^x-e(1+x)/2$. They satisfy the left and right Dirichlet boundary conditions, and $W=u{v}^{\prime }-u^{\prime }v=x{e}^x$. The displayed kernel is valid for $\xi \ne 0$ and has unit jump in its first $x$ derivative. Its $1/\xi$ behaviour prevents unrestricted integration through zero. For smooth forcing $f(x)=xq(x)$, the product $G(x,\xi )f(\xi )$ has a removable singularity, and its integral gives the unique solution extending with two continuous derivatives through zero. That matching condition matters: merely requiring one continuous derivative allows additional piecewise homogeneous solutions. A smooth forcing with $f(0)\ne 0$ cannot have a solution with two continuous derivatives there, as expansion of $x{y}^{\prime \prime }-(1+x)y^{\prime }+y=xf$ shows.