I found this characteristic equation $r(r-1)^2=1$ in my homework Euler differential equation problem (page 669, again the book), $x^{3} y'''+xy'-y=0$, when I misread the problem as
$$x^{3} y'''+xy''-y=0$$
so this beast has characteristic equation that looks nightmarish at least here in WA. I am not sure whether I am missing some smart substitution to proceed, how would you solve the polynomial?
Please, keep the focus on the polynomial. Thanks for pointing out the err but I would like to know how to solve this kind of polynomials.