= Solution
In the <Maxwell-Boltzmann velocity distribution>, each Cartesian component has mean-square velocity $k_BT_e/m_e$. Thus $\langle v^2\rangle=3k_BT_e/m_e$. The original PDF calls this quantity an rms velocity, but the rms speed is its square root. Also, its $n_\gamma$ must mean the dimensionless mean <occupation number> $n$, since its equilibrium value is a <Planck photon distribution>; it is not the frequency-integrated <photon number density>.
Take $x=h\nu/(k_BT_\gamma)$ at a fixed local radiation temperature. Because $x$ is proportional to <frequency>,
$$
\nu^2\partial_\nu^2n+4\nu\partial_\nu n
=x^2\partial_x^2n+4x\partial_xn
=x^{-2}\partial_x(x^4\partial_xn).
$$
The hot-electron <Kompaneets equation> therefore becomes
$$
\boxed{\frac{\partial n}{\partial y}=x^{-2}\frac{\partial}{\partial x}\left(x^4\frac{\partial n}{\partial x}\right),\qquad
dy=\frac{k_BT_e}{m_ec^2}n_e\sigma_Tc\,dt.}
$$
For traversal length $d\ell=c\,dt$, the <Compton y parameter> is
$$
\boxed{y=\frac{\sigma_T}{m_ec^2}\int n_ek_BT_e\,d\ell
=\frac{\sigma_T}{m_ec^2}\int P_e\,d\ell.}
$$
It measures electron <pressure> integrated along the <line of sight>, or the <Thomson scattering> <optical depth> weighted by $k_BT_e/(m_ec^2)$. In this hot-electron limit the mean fractional <energy> gain is approximately $4y$. The derivation assumes nonrelativistic electrons, Thomson-limit photon <energies> and $T_e\gg T_\gamma$. The original PDF uses $h\nu$, as required for ordinary <frequency>; the converted TeX incorrectly substitutes $\hbar\nu$.
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