= Solution
Set $t=\log x$, so $0<t<\pi$, and write $y(x)=x^{-2}w(t)$. Direct differentiation of this <Cauchy-Euler equation> gives
$$
x^2y''+5xy'+y=x^{-2}(w''-3w).
$$
The <eigenvalue> equation is therefore $-w''=(\lambda-3)w$ with $w(0)=w(\pi)=0$. For $\lambda-3\leq0$, the linear/hyperbolic solutions cannot satisfy both endpoint conditions except trivially. For $\lambda-3>0$, the sine solution vanishes at both endpoints exactly when its frequency is a positive <integer>. Thus
$$
\boxed{\lambda_k=k^2+3,\qquad y_k(x)=x^{-2}\sin(k\log x),\qquad k=1,2,\ldots.}
$$
Each associated <eigenspace> is one-dimensional.
\b[The displayed differential operator is not <self-adjoint> for the unweighted <inner product>.] Let $A=x^2\partial_x^2+5x\partial_x+1$. <Integration by parts> on compactly supported smooth functions gives its <formal adjoint>
$$
A^*f=(x^2f)''-(5xf)'+f=x^2f''-xf'-2f,
$$
which differs from $A$. One can see an explicit failure of symmetry without boundary terms: take a nonzero real compactly supported smooth $f$ in the interval and $g=xf$. Then
$$
\langle f,Ag\rangle-\langle Af,g\rangle=3\int_1^{e^\pi}x(fg'-f'g)\,dx=3\int_1^{e^\pi}xf^2\,dx>0.
$$
Both <test functions> obey the Dirichlet endpoint conditions. Changing the sign of $A$ for the eigenproblem does not repair this failure. A different, weighted <inner product> with weight $x^3$ does give the <Sturm-Liouville form> $x^3Ay=(x^5y')'+x^3y$, explaining why a real discrete <spectrum> is nevertheless natural.
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