  Fourth order problems, with the differential equation
$y^{(4)}-(gy')'=\lambda^2y$, where $g\in C^1[0,a]$ and $a>0$, occur in
engineering on stability of elastic rods. They occur as well in aeronautics to
describe the stability of a flexible missile. Fourth order Birkhoff regular
problems with the differential equation $y^{(4)}-(gy')'=\lambda^2y$ and
eigenvalue dependent boundary conditions are considered. These problems have
quadratic operator representations with non self-adjoint operators. The first
four terms of the asymptotics of the eigenvalues of the problems as well as
those of the eigenvalues of the problem describing the stability of a flexible
missile are evaluated explicitly.
