By means of the two-dimensional basic equations of transversely isotropic magneto-electro-elastic media and the strict differential operator theorem, the general solution in the case of distinct eigenvalues is derived, in which all mechanical, electric and magnetic quantities are expressed in four harmonic displacement functions. Based on this general solution in the case of distinct eigenvalues, a series of problems is solved by the trial-and-error method, including magneto-electro-elastic rectangular beam under uniform tension, electric displacement and magnetic induction, pure shearing and pure bending, cantilever beam with point force, point charge or point current at free end, and cantilever beam subjected to uniformly distributed loads. Analytical solutions to various problems are obtained.