= Solution
Let $\zeta=z/H(t)$ label a fluid element and use the <homologous vertical motion of an astrophysical disk>
$$
\rho=\frac\Sigma H\widetilde\rho(\zeta),
\qquad
p=\frac{P(t)}H\widetilde p(\zeta),
\qquad
\mathbf u=-Sx\mathbf e_y+\dot H\zeta\mathbf e_z.
$$
Because $D\zeta/Dt=0$ and $\nabla\mathbin\cdot\mathbf u=\dot H/H$, this ansatz satisfies <mass conservation>. The pressure equation for an <adiabatic process> gives
$$
\frac{\dot P}{P}+(\gamma-1)\frac{\dot H}{H}=0,
$$
hence
$$
\boxed{PH^{\gamma-1}=\text{constant},
\qquad P\mathrel\propto H^{1-\gamma}}.
$$
The vertical acceleration is $D u_z/Dt=\ddot H\zeta$. Using the dimensionless hydrostatic profiles from part c, the pressure force is $(P/\Sigma H)\zeta$, and the vertical momentum equation reduces to
$$
\boxed{\ddot H+\Omega_z^2H=CH^{-\gamma}},
$$
where $C$ is constant.
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