= Dirichlet Green kernel for an Euler operator with exponents one and minus four
{c}
{title2=$G(x,\xi)=\xi^4x_<\left(x_>-x_>^{-4}\right)/5$}
For $L=d^2/dx^2+4x^{-1}d/dx-4x^{-2}$ 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 $L_xG=\delta(x-\xi)$. For bounded forcing $f$, $y(x)=\int_0^1G(x,\xi)f(\xi)d\xi$ 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.
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