= Dirichlet Green function for a radial power operator on an interval
{c}
{title2=$L=u''+(n+1)u'/x$}
Let $0<a<b$ and impose zero <Dirichlet boundary conditions> at $a,b$. With the convention $L_xG(x,s)=\delta(x-s)$ for the <Dirac delta distribution>, the <Green's function> is continuous and its first <derivative> has jump one. For $n\ne0$ choose $\ell(x)=x^{-n}-a^{-n}$ and $r(x)=x^{-n}-b^{-n}$. Their <Wronskian> is $n(a^{-n}-b^{-n})s^{-n-1}$, so
$$
G(x,s)=\frac{s^{n+1}(x_<^{-n}-a^{-n})(x_>^{-n}-b^{-n})}{n(a^{-n}-b^{-n})},\qquad x_<=\min(x,s),\quad x_>=\max(x,s).
$$
For $n=0$, take the continuous limit or use $\ell=\log(x/a)$ and $r=\log(x/b)$ to obtain
$$
G(x,s)=\frac{s\log(x_</a)\log(x_>/b)}{\log(b/a)}.
$$
The solution is $u(x)=\int_a^bG(x,s)f(s)\,ds$. These kernels use ordinary measure $ds$. The factor $s^{n+1}$ comes from the nonconstant <Wronskian> and must not be discarded by confusing the original operator with its multiplied <divergence form> $(x^{n+1}u')'$.
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