= Solution
Define the <weighted inner product> $\langle u,v\rangle_w=\int_a^bw(x)\overline{u(x)}v(x)dx$. For the stated operator $L=-(pD)' -q$, integration by parts gives the <Lagrange identity for a Sturm-Liouville operator>
$$
\int_a^b[\overline uLv-\overline{Lu}v]dx=[p(\overline{u'}v-\overline u v')]_a^b.
$$
The endpoint terms vanish for <Dirichlet boundary conditions>. Taking $u=v$ proves each eigenvalue is real, since $w>0$ makes the squared weighted norm positive. For eigenfunctions $u_m,u_n$ of distinct eigenvalues, the same identity gives $(\lambda_n-\lambda_m)\langle u_m,u_n\rangle_w=0$, proving their orthogonality.
For the specific equation, divide by $x^2$ to obtain $-y''=\lambda x^{-2}y$, so its weight is $w=x^{-2}$, with $p=1,q=0$. Put $v=\ln x$, $\ell=\ln2$ and $Y(v)=y(e^v)$. The <logarithmic change of variable in a Sturm-Liouville problem> gives $x^2y''=Y''-Y'$. Remove the first derivative with $Y=e^{v/2}Z$; the equation becomes $Z''+(\lambda-1/4)Z=0$, with $Z(0)=Z(\ell)=0$.
The <Dirichlet spectrum with inverse-square weight> is therefore
$$
\boxed{\lambda_n=\frac14+\left(\frac{n\pi}{\ln2}\right)^2,\qquad y_n(x)=\sqrt{\frac2{\ln2}}\sqrt x\sin\frac{n\pi\ln x}{\ln2},\quad n\geq1.}
$$
Indeed $\int_1^2x^{-2}y_my_n dx=(2/\ell)\int_0^\ell\sin(m\pi v/\ell)\sin(n\pi v/\ell)dv=\delta_{mn}$. Values $\lambda\leq1/4$ give a linear or hyperbolic transformed solution and no nonzero solution vanishing at both ends.
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