Eigenvalue singular problem of factorized fourth-order self-adjoint differential equations

This paper deals with the eigenvalue singular problem of the spectral type differential equations of the fourth-order self-adjoint differential operators1(1-x)p(1+x)qd2dx2(1-x)p+2(1+x)q+2d2ydx2-μy=0,y(-1),y(1)finite where, p⩾1,q⩾1. It has been reported in the literature that spectral type differential equations above have general solutions in terms of hypergeometric functions. In this work it is showed that the general solution in terms of hypergeometric functions reduces to Jacobi orthogonal polynomials (as eigenfunctions) in the case of eigenvalue singular problem. The corresponding eigenvalues are found. As application, the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform singular structure are reported.