This paper analyzes the downlink achievable spectral efficiency of an orthogonal frequency-division multiplexing (OFDM) cooperative cognitive network with a non-linear relay. The analysis is carried out subject to the power amplifier's (PA) non-linear effect on the relay node which operates in the amplify-and-forward (AF) mode. Specifically, an analytical expression for the power spectral density (PSD) of relay output in terms of its source power is derived. By using the obtained PSD, the power of relay's PA output is derived for each subcarrier and the 1 dB compression point is determined. Then, the adjacent channel power (ACP) of each subcarrier is analytically derived in terms of secondary user (SU) source input power. Next, the signal to interference and noise ratio (SINR) of each subcarrier at destination (SU receiver) is calculated by considering the non-linear effect of PA at the relay. Having the SINRs of all subcarriers, a constrained optimization problem on the source's input power is formulated in which the achievable spectral efficiency is the objective function and all ACPs being less than the interference temperature limit are its constraints. Finally, we perform some simulations and show that the numerical results are consistent with the analytical findings.