On the computing of the eigenvalues of the Orr–Sommerfeld problem

The paper deals with the Orr–Sommerfeld problem
\begin{align*} {} & \{(iR)^{-1}M^2-\alpha[q(x)M-q''(x)]\}y=-\lambda My,\\ &y(\pm 1)=y'(\pm1)=0, \end{align*}
where $M=d^2/dx^2-\alpha^2$, $q(x)$ is the velocity profile, $R$ and $\alpha$ are Reynolds and wave numbers, respectively. We approve the Galerkin method to compute the eigenvalues of this problem provided that the basis for the method consists of the eigenfunctions of the operator $M^2$.