A family of isospectral fourth order Sturm-Liouville problems

In this paper, we consider the class of fourth order Sturm Liouville equation of the form y^{(4)} (z) − 2(q(z)y′)′ + (q^ 2 (z) − q′′(z))y(z) = λ^2y(z), 0 ≤ z ≤ L, with boundary conditions y(z) = y′′(z) = 0 at z = 0,L. We prove that this class is equivalent to a second order Sturm Liouville problem. Using Darboux Lemma we obtain the closed form of fourth order Sturm Liouville equations that is isospectral to a given one