On the Solutions of Multi-dimensional Clairaut Equation with Multi-homogeneous Function of the Derivatives

The analysis of the solutions of Clairaut equation with an arbitrary number of independent variables is completed. It is assumed that the function of the derivatives, which is part of the equation is multi-homogeneous. This means that the set of function arguments can be represented as the union of subsets, and the function is homogeneous on each of these subsets. We consider solutions of equations depending on linear combinations of the original variables, each of which contains only a certain subset of variables. Original equation is transformed to a reduced one, which can be solved by separation of variables. It is shown that the reduced equation has solutions in the form of arbitrary homogeneous functions with index of homogeneity 1 and solutions in the form of some generalized polynomials.