= Solution
The useful <change of independent variable in a second-order ODE> is $t=\log x$, followed by $y(x)=xv(t)$. The <chain rule> gives
$$
y'=v+v_t,\qquad y''=x^{-1}(v_t+v_{tt}),\qquad Ly=xv_{tt}.
$$
Thus the <eigenvalue> equation becomes $v_{tt}=\lambda v$ on $0<t<1$, and the endpoint values become $v(0)=v(1)=0$. At the specified <eigenvalues>,
$$
v(t)=A\cos(k\pi t)+B\sin(k\pi t).
$$
The first endpoint forces $A=0$; the second holds for every positive integer $k$. Choosing a convenient normalization yields
$$
\boxed{y_k(x)=x\sin(k\pi\log x),\qquad k=1,2,\ldots.}
$$
Each nonzero constant multiple is the same <eigenfunction> up to normalization. This <logarithmic conjugacy of a repeated-root Euler operator> turns the <Cauchy-Euler equation> into the familiar <Fourier sine series> problem on a unit <interval>.
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