= Solution
Multiplying by $x^2$ makes the first two terms a total <derivative>:
$$
x(x-1)y''+(2x-1)y'=\left[x(x-1)y'\right]'.
$$
The resulting <Sturm-Liouville form> is
$$
\boxed{-(p y')'+l(l+1)y=\lambda w y,\qquad p=x(x-1),\quad w=x^2.}
$$
Equivalently, an integrating multiplier $\mu$ must satisfy $(\mu(1-1/x))'=\mu(2/x-1/x^2)$, which reduces to $\mu'/\mu=2/x$ and gives $\mu=x^2$ up to a constant. The coefficient $p$ is positive for $1<x<R$ and vanishes linearly at the left endpoint; the weight $w$ is positive throughout. The boundedness condition removes the logarithmic local branch, giving the <no-log boundary condition at a linearly vanishing Sturm-Liouville endpoint>.
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