= Logarithmic conjugacy of a repeated-root Euler operator
{title2=$x^2D^2-xD+1=x\,(d^2/dt^2)\,x^{-1},\quad t=\log x$}
For $x>0$, write $t=\log x$ and $y=xv(t)$. The <chain rule> gives $y'=v+v_t$, $y''=x^{-1}(v_t+v_{tt})$, and therefore
$$
(x^2D^2-xD+1)y=xv_{tt}.
$$
On $1<x<e^\ell$ with homogeneous <Dirichlet boundary conditions>, the <eigenfunctions> are $x\sin(n\pi\log x/\ell)$ and the <eigenvalues> are $-(n\pi/\ell)^2$. The <weighted inner product> with weight $x^{-3}$ becomes $\int_0^\ell\overline v w\,dt$, so their squared <norms> are $\ell/2$ and their <orthogonality> is exactly that of the <Fourier sine series>. The weight is the factor that makes this conjugacy preserve the <inner product>.
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