Frobenius & Special Functions · Superchallenge problem

The equation $x^2y'' - 2xy' + (\lambda x - x^2)y = 0$ has indicial roots $0$ and $3$ at $x = 0$, so in general its second solution contains a $\ln x$ term. For exactly which real $\lambda$ does the root $r = 0$ nevertheless give a genuine Frobenius series (no logarithm)? Compute the sum of the squares of all such $\lambda$.