= Solution
The first equation is vertical <hydrostatic equilibrium>: the upward <pressure gradient> balances the central <mass>'s vertical gravitational force. For $|z|\ll r$, the point-mass acceleration is $-GMz/(r^2+z^2)^{3/2}\simeq-\Omega_K^2z$, where <Keplerian rotation> gives $\Omega_K^2=GM/r^3$. The disc's <self-gravity> and vertical inertia are neglected.
The second equation is local thermal balance. The divergence of the vertical <radiative flux> equals <viscous dissipation>. In the convention of this <alpha disk>, $\nu=\alpha p/(\rho\Omega)$, and the dominant <Keplerian shear> has $r\,d\Omega/dr=-3\Omega/2$. Therefore
$$
q^+=\rho\nu\left(r\frac{d\Omega}{dr}\right)^2=\frac94\alpha\Omega p=\frac{\partial F}{\partial z}.
$$
The <radiative flux> increases from zero at the symmetric midplane to a positive outgoing value at the upper surface. Radial heat transport, thermal <advection> and rapid storage of thermal <energy> are excluded from this local balance.
The third equation is <radiative diffusion>, equivalently
$$
F=-\frac{16\sigma T^3}{3\kappa\rho}\frac{\partial T}{\partial z}.
$$
It applies to optically thick material, with $\kappa$ the opacity per unit <mass> and $\sigma$ the <Stefan-Boltzmann constant>. Here constant <electron-scattering opacity> represents <Thomson scattering>; the formula uses a diffusive, nearly thermal radiation field. Positive upward <flux> requires <temperature> to decrease upwards.
The final equation is the <ideal gas> <pressure> law, with <mean molecular weight> $\mu_m$ in proton-mass units. Writing $\mathcal R=k/(\mu_m m_p)$ gives $p=\mathcal R\rho T$. The <pressure> is <gas pressure> because <radiation pressure> is assumed negligible. \b[The four equations express vertical support, local heating balance, diffusive cooling and the gas equation of state.]
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