Apparently the first closed-form solution of vibrating inhomogeneous beam with a tip mass

Whereas there are numerous papers on the free vibrations of beams of uniform or non-uniform cross-section carrying concentrated masses, the problem does not lend itself to the closed-form solution. Here such a solution is reported, apparently for the first time. The solution originally derived for the inhomogeneous beam without a concentrated mass is generalized to include a tip mass. The semi-inverse method is utilized, to achieve this goal.