Forced axisymmetric density mode with vortensity (${\ddot{\Sigma }}^{\prime }+{\omega }_k^2{\Sigma }^{\prime }=-2\Omega {\Sigma }_0^2{q}^{\prime }$)

For a radial mode with constant nonzero wavenumber in an inviscid razor-thin disc, eliminate velocity using continuity and the conserved linearized vortensity. The resulting oscillator has ${\omega }_k^2={\kappa }^2+c_s^2{k}^2-2\pi G{\Sigma }_0|k|$. The vortensity gives a constant forcing and hence the stationary balanced component when ${\omega }_k^2>0$. It is lost by an ansatz that divides every amplitude equation by a nonzero frequency. Negative squared frequency gives exponential instability; zero frequency allows secular growth.