= Solution
With the <spontaneous emission> term suppressed, $j_\nu=0$. A uniform <absorption coefficient> over length $\ell$ gives directly
$$
\boxed{I_\nu=I_\nu^0e^{-\tau_\nu},\qquad \tau_\nu=\ell\alpha_\nu.}
$$
The normalized <specific intensity> deficit is unchanged by conversion between <wavelength> and <frequency>, since both numerator and denominator acquire the same <Jacobian determinant>. The <Boltzmann distribution> and the <Einstein coefficients> give $n_2B_{21}/(n_1B_{12})=e^{-h\nu_{12}/(k_BT_e)}\ll1$. Consequently, in the <optically thin> limit,
$$
W=\int(1-e^{-\tau_\nu})\frac{c}{\nu^2}\,d\nu
\simeq\frac{hN_1B_{12}}{\nu_{12}}\int\varphi(\nu)d\nu
=\boxed{\frac{hB_{12}N_1}{\nu_{12}}}.
$$
Here $N_1=n_1\ell$ is the <column density>, and $\nu\simeq\nu_{12}$ has been used in slowly varying factors. This linear relation between <equivalent width> and <column density> fails once the central <optical depth> becomes appreciable.
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