= Forced axisymmetric density mode with vortensity
{title2=$\ddot\Sigma'+\omega_k^2\Sigma'=-2\Omega\Sigma_0^2q'$}
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 $\omega_k^2=\kappa^2+c_s^2k^2-2\pi G\Sigma_0|k|$. 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.
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