A general fourth order ordinary differential equation with solution in terms of Bessel functions: theory and engineering applications

Bessel functions are the solutions of Bessel's differential equation occur in numerous problems of mathematical physics and engineering, especially in cylindrical and spherical coordinates. The problem of transverse vibration of non-uniform beams and determination of their natural frequencies has been vastly studied by engineers and mathematicians. This problem leads to a fourth order ordinary differential equation which generally has not a closed-form solution. The general form of a fourth order ordinary differential equation is derived. The solution of which is analogous to Bessel's differential equation, may be expressed in terms of Bessel functions. This fourth order differential equation along with boundary conditions represents an eigenvalue problem. The application of this differential equation in transverse vibration of some particular cases of non-uniform Euler-Bernoulli beams is demonstrated by recovering available solutions to particular cases.