  We consider a class of Hill equations where the periodic coefficient is the
squared solution of some Duffing equation plus a constant. We study the
stability of the trivial solution of this Hill equation and we show that a
criterion due to Burdina (V.I. Burdina, Boundedness of solutions of a system of
differential equations) is very helpful for this analysis. In some cases, we
are also able to determine exact solutions in terms of Jacobi elliptic
functions. Overall, we obtain a fairly complete picture of the stability and
instability regions. These results are then used to study the stability of
nonlinear modes in some beam equations.
