= Solution
In logarithmic coordinates the <eigenvalue problem> becomes $-Y''=(\lambda+1)Y$. A nonzero function with zero values at both ends requires $\lambda+1>0$: multiplying by $Y^*$ and integrating gives $(\lambda+1)\int_0^1|Y|^2dt=\int_0^1|Y'|^2dt>0$. Write $\kappa=\sqrt{\lambda+1}$. The first endpoint gives $Y=A\sin\kappa t$, and the second requires $\kappa=n\pi$, $n=1,2,\ldots$.
For <vector normalization>, $\int_1^e|y|^2dx/x=|A|^2\int_0^1\sin^2(n\pi t)dt=|A|^2/2$. A positive real normalization constant therefore gives
$$
\boxed{\lambda_n=n^2\pi^2-1,\qquad y_n(x)=\sqrt2\sin(n\pi\ln x),\quad n\geq1.}
$$
The sine <orthogonality> identity also gives $\int_1^ey_m y_n\,dx/x=\delta_{mn}$. Completeness is the usual completeness of the <Fourier sine series> on $(0,1)$ under the logarithmic coordinate transformation.
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