= Solution
Use a <logarithmic change of variable in a Sturm-Liouville problem>: set $t=\log x$ and $Y(t)=y(e^t)$. The <chain rule> gives
$$
y'=\frac{Y'}x,\qquad y''=\frac{Y''-Y'}{x^2},\qquad \mathcal Ly=-Y''-Y.
$$
The endpoint conditions become $Y(0)=Y(1)=0$. Multiplying $-Y''=(\lambda+1)Y$ by $Y^*$ and integrating shows that $\lambda+1>0$ for a nonzero solution. Put $\mu=\sqrt{\lambda+1}$. The first endpoint selects $Y=A\sin(\mu t)$ and the second requires $\mu=n\pi$, $n\geq1$. Hence
$$
\boxed{\lambda_n=n^2\pi^2-1,\qquad y_n(x)=\sqrt2\sin(n\pi\log x),\quad n=1,2,\ldots.}
$$
These <eigenfunctions> are orthonormal in the specified <weighted inner product>, because $w(x)dx=dx/x=dt$ and
$$
\int_1^e\frac{y_m(x)y_n(x)}x\,dx
=2\int_0^1\sin(m\pi t)\sin(n\pi t)\,dt=\delta_{mn}.
$$
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