= Solution
The one-group particle balance with a prescribed external volume source $S$ can be written
$$
\frac1v\frac{\partial\phi}{\partial t}
=-\nabla\cdot\mathbf J-\Sigma_a\phi+\nu\Sigma_f\phi+S,
\qquad
\mathbf J=-D\nabla\phi.
$$
Here $\phi$ is scalar <neutron flux>, $D$ is the <neutron diffusion coefficient>, and $\Sigma_a,\Sigma_f$ are <macroscopic cross sections>. The terms describe accumulation, current divergence, absorption, <nuclear fission> production and external input.
For a <steady state>, set $\partial_t\phi=0$; source-free means $S=0$. In a homogeneous medium $D$ is spatially constant, so substitution gives
$$
D\nabla^2\phi+(\nu\Sigma_f-\Sigma_a)\phi=0.
$$
The supplied notation defines $\eta$ as the effective production per absorption, $\eta=\nu\Sigma_f/\Sigma_a$. Therefore
$$
\boxed{\nabla^2\phi+\frac{(\eta-1)\Sigma_a}{D}\phi=0.}
$$
This identification uses the absorption accounting for the homogeneous model; a fuel-only reproduction factor with other absorbers excluded would not automatically equal this $\eta$. The derivation assumes one prescribed energy group, constant effective coefficients and the <diffusion approximation>; all neutron production included here is represented by the steady <nuclear fission> term.
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