On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions

In the theory of ordinary differential equations, the Clairaut equation is well known. This equation is a non-linear differential equation unresolved with respect to the derivative. Finding the general solution of the Clairaut equation is described in detail in the literature and is known to be a family of integral lines. However, along with the general solution, for such equations there exists a singular (special) solution representing the envelope of the given family of integral lines. Note that the singular solution of the Clairaut equation is of particular interest in a number of applied problems.
In addition to the ordinary Clairaut differential equation, a differential equation of the first order in partial derivatives of the Clairaut type is known. This equation is a multidimensional generalization of the ordinary differential Clairaut equation, in the case when the sought function depends on many variables. The problem of finding a general solution for partial differential equations of the Clairaut is known to be. It is known that the complete integral of the equation is a family of integral (hyper) planes. In addition to the general solution, there may be partial solutions, and, in some cases, it is possible to find a singular solution. Generally speaking, there is no general algorithm for finding a singular solution, since the problem is reduced to solving a system of nonlinear algebraic equations.
The article is devoted to the problem of finding a singular solution of Clairaut type differential equation in partial derivatives for the particular choice of a function from the derivatives in the right-hand side. The work is organized as follows. The introduction provides a brief overview of some of the current results relating to the study of Clairaut-type equations in field theory and classical mechanics. The first part provides general information about differential equations of the Clairaut-type in partial derivatives and the structure of its general solution. In the main part of the paper, we discuss the method for finding singular solutions of the Clairaut-type equations. The main result of the work is to find singular solutions of equations containing power and exponential functions.