I am trying to solve this eigenvalue problem using the function eig/eigs, and if made easier, the chebyshev numerical method as well.

Trying to solve for the velocity vector field using U,V, and W, which are velocity amplitudes as functions of radial position, r, only. Viscosity nu is known, A and B are known, r = ra < r < rb. alpha and sigma are not known, but are complex numbers, and are the eigenvalues . See equations below. In specific, I am trying to solve the rz-momentum eqn at the bottom, the others are shown just in case. Boundary values are U=V=W=0 at r=ra and r=rb