Turn 2nd order ODE in to Sturm-Liouville Form How Do I turn
$$(x+1)y''-xy'+y=0$$
into Sturm-Liouville Form?

what you need is an $a$ so that you have $$a(x+1)y''-axy' = (a(x+1)y')'=a(x+1)y''+(a'(x+1)+a)y'$$ therefore choose $a$ to satisfy $$a'(x+1) +a = -ax \to \frac {a'}a = -1\to a= e^{-x} $$ with that we have $$0= e^{-x}(x+1)y''-e^{-x}xy' +e^{-x}y \to \left(e^{-x}(x+1)y'\right)'+e^{-x}y$$ in a self adjoint form.
