Factorization of self-adjoint ordinary differential equations

This paper deals with the factorization of self-adjoint differential operators L(2n)=1ρdndxnρβndndxn, and their spectral type differential equations. Sufficient conditions of factorization are reported. A large class of differential operators and equations that can be factorized is obtained. The factorizations of fourth- and sixth-order operators and equations are explicitly given. A particular fourth-order spectral type differential equation in which ρ(x)=(1-x)p(1+x)qρ(x)=(1-x)p(1+x)q, p⩾1,q⩾1, is considered. Its general solution is obtained in terms of hypergeometric functions. As application, the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform structure are found.