Classes of exact solutions for several static and dynamic problems of non- uniform beams

In this paper, an analytical procedure for solving several static and dynamic problems of non-uniform<br />beams is proposed. It is shown that the governing differential equations for several stability, free<br />vibration and static problems of non-uniform beams can be written in the from of a unified self-conjugate<br />differential equation of the second-order. There are two functions in the unified equation, unlike most<br />previous researches dealing with this problem, one of the functions is selected as an arbitrary expression<br />in this paper, while the other one is expressed as a functional relation with the arbitrary function. Using<br />appropriate functional transformation, the self-conjugate equation is reduced to Bessel