I need help numerically evaluating the characteristic polynomial of a matrix and solving for its roots. This question relates to Eq. 3.111a of $\left[1\right]$ Munjal, M.L. (2014); "Acoustics of Ducts and Mufflers", $2^{nd}$ ed., Wiley
A set of second-order equations has been transformed into a set of four simultaneous first-order equations: $$ \begin{bmatrix} -1 & 0 & D & 0 \\ 0& -1 & 0 & D \\ D & 0 & \alpha_1D+\alpha_2 & \alpha_3D+\alpha_4 \\ 0 & D & \alpha_5D+\alpha_6 & \alpha_7D+\alpha_8\end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \tag 1$$ or $$\begin{bmatrix} \Delta \end{bmatrix} \begin{Bmatrix} y\end{Bmatrix} = \begin{Bmatrix} 0\end{Bmatrix}$$ where $$p'_1(z)=y_1\text{, }p'_2(z)=y_2 \text{, }p_1(z)=y_3\text{, }p_2(z)=y_4\text{ and } D= \frac{d} {dz}$$ All $\alpha_i$ in $(1)$ are known complex numbers. Equation $(1)$ is then transformed to the principal variables $\Gamma_i$ as: $$ \begin{bmatrix} D-\beta_1 & 0 & 0 & 0 \\ 0 & D - \beta_2 & 0 & 0 \\ 0 & 0 & D-\beta_3 & 0 \\ 0 & 0 & 0 & D-\beta_4\end{bmatrix} \begin{bmatrix} \Gamma_1 \\ \Gamma_2 \\ \Gamma_3 \\ \Gamma_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \tag 2$$
The principal state variable $\Gamma_i$ are related to the variable $y_i$ through the eigenmatrix $\left[\psi\right]$ as $$\begin{Bmatrix} y \end{Bmatrix} = \begin{bmatrix}\psi\end{bmatrix}\begin{Bmatrix}\Gamma \end{Bmatrix}$$ To quote from $\left[1\right]$: "The $\beta_i$'s are the zeros of the characteristic polynomial $\begin{vmatrix}\Delta\end{vmatrix}$."
And therein lies my problem - I have no idea how to deal with the $D$ in $(1)$ while evaluating the characteristic polynomial numerically. I tried forming the polynomial $CP=det\left(t\cdot I-\begin{vmatrix}\Delta\end{vmatrix}\right)$ to check whether $D$ cancels out by sheer luck - but it doesn't (unless I erred there).
How do I go about calculating the $\beta_i$'s in this case, given that neither the principal state variable $\Gamma_i$ nor the variable $y_i$ is known a priori? Your suggestions are greatly appreciated.