This work continues our studies in the properties of the homogeneous Euler' s functions that can be used in the synthesis of electric and magnetic fields for electron and ion-optical systems to carry out spectrographic recording mode. A generalization of a functional general equation for homogeneous functions has been considered. This equation corresponds to linear functional relations with a minimal-sized matrix. A general solution of the obtained functional equation was found assuming of differentiability of the functions in question. The resulting systems of functions were termed mutually homogeneous functions by analogy with the homogeneous Euler's functions and the associated homogeneous Gel’fand’s functions.