A Clairaut-type differential equation in partial derivatives with a power function

The article is aimed at the study of Clairaut-type differential equations in partial derivatives of the first order with a special right-hand side. Finding a general solution for an ordinary Clairaut differential equation is not difficult, and it is described in detail in courses on the theory of ordinary differential equations. In addition to the general solution, which involves a family of integral lines, there is a special solution for an ordinary Clairaut differential equation, represented by the envelope of this family. Within the framework of the theory of differential equations in partial derivatives there are Clairaut-type differential equations, which are multidimensional generalizations of an ordinary Clairaut differential equation. Note that for a Clairaut-type partial differential equation there is no always a special solution. The article is devoted to the problem of the description of a particular solution for Clairaut-type equations, which right side has the form of a power function of the product of n factors.