 Solution

The constant term of the real cubic dispersion relation isA negative $c$ guarantees a positive real root because the cubic tends to $+\infty$ as $\lambda \to +\infty$.

If $\beta <0$, then $c<0$ at sufficiently small positive $k$, producing a viscous instability of an accretion disk. This is the local form of the global thin-disk diffusion equationSince $d(\nu \Sigma )/d\Sigma <0$, the effective diffusion reverses sign and amplifies surface-density variations.

If $\beta >0$, instability is still possible when the quadratic expression in parentheses is negative. Its minimum occurs at $k=\pi G{\Sigma }_0/v_s^2$, so the condition isorThis is a secular gravitational instability of an astrophysical disk: viscosity allows self-gravity to overcome rotational support even in part of the range that is stable by the inviscid $Q<1$ criterion.