Factorization of self-adjoint ordinary differential equations

Keyword: Factorization method Self-adjoint differential equations Eigenvalue problems This paper deals with the factorization of self-adjoint differential operators Lð2nÞ 1⁄4 1 q d n dx qb d n dx , 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 fourthand sixth-order operators and equations are explicitly given. A particular fourth-order spectral type differential equation in which qðxÞ 1⁄4 ð1 xÞð1þ xÞ, p P 1; q P 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. 2013 Elsevier Inc. All rights reserved.