Suppose we have a system of equations: \begin{gather*} \left(\begin{array}{cccc} c\partial_t +\partial_x & \alpha & 0 &0 \\ N_0 & \partial_t & -\sigma_2 \frac{Q_0}{N_0} & \sigma_2 \omega_0 \\ 0 & \sigma_1 \sigma_2 \frac{Q_0}{N_0} & \partial_t & 0 \\ 0& -\omega_0 & 0 &\partial_t\end{array}\right)\left(\begin{array}{c}\delta E\\\delta P \\\delta N \\\delta Q\end{array}\right) = 0, \tag1 \end{gather*} We now assume perturbations are plane waves of the form $δN ∝ \cos(κx + ωt)$ for all four fields. By a direct substitution into (1), we obtain \begin{gather*} -c \omega ^3 - \kappa \omega^2 +(c \omega_0 \frac{\sigma_2 N_0^2 +\sigma_1 Q_0^2}{N_0^2}-\alpha N_0)\omega +\kappa \omega_0 \frac{\sigma_2 N_0^2 +\sigma_1 Q_0^2}{N_0^2} = 0. \tag2 \end{gather*}
How to find the relation (2)? By equating the coefficients of $\cos(κx + ωt)$ and $\sin(κx + ωt)$ to zero? I tried it but couldn't find it.