Dirichlet Green kernel for an Euler operator with exponents one and minus four ($G(x,\xi )={\xi }^4{x}_{<}\left(x_{>}-x_{>}^{-4}\right)/5$)

For $L=d^2/d{x}^2+4x^{-1}d/dx-4x^{-2}$ on $(0,1)$, regularity at zero selects the homogeneous solution $u=x$, while the right Dirichlet boundary condition selects $v=x^{-4}-x$. Their Wronskian is $-5x^{-4}$. Thus the displayed Green function, with $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$, is continuous, has unit first-derivative jump and satisfies $L_{x}G=\delta (x-\xi )$. For bounded forcing $f$, $y(x)={\int }_0^{1}G(x,\xi )f(\xi )d\xi$ is the unique solution continuous at zero with zero endpoint values. The singular homogeneous branch $x^{-4}$ is excluded by that regularity. The kernel uses ordinary source measure $d\xi$, not the measure for the multiplied self-adjoint operator.