Solution of Clairaut differential equations in partial derivatives with logarithmic function

Background. Special solutions of partial differential Clairaut-type equations are of particular interest in applied problems. In recent papers, the connection of a particular solution of the Clairaut-type equation and the effective action in quantum field theory with compound operators has been discussed. The purpose of this paper is to describe a method for finding particular solutions for equations of the Clairaut-type, the right-hand side of which has the form of a logarithmic function of the product - independent variables. Materials and methods. A procedure has been proposed for finding a special solution of a Clairaut-type equation for the case when the function has the form of a logarithmic function. The basic idea is to find not the functions, but the expressions $a^i z_i$ and $x^i z_i$. This method can be used to find special solutions of equations of the Clairaut-type for some functions in which this structure is preserved. Results. In this paper, we consider the problem of finding a special solution of a partial differential Clairaut-type equation for the case when the function of the derivatives is the logarithm of the product of independent variables. The case when all degrees of derivatives under the logarithm sign have the same value is discussed separately, and the derivation of a special solution for an equation in the case of arbitrary different degrees is also discussed in detail. The special solutions obtained in this work were calculated for an arbitrary number of variables and represent the main result of the work. Conclusions. We studied differential Clairaut-type equations in partial derivatives. This type of equations is a nonlinear partial differential equation and is a generalization of the well-known ordinary Clairaut differential equation to the case when the desired function depends on many variables. The method of finding a general solution for equations of this type is described in detail in the literature. However, there is no general method for finding a particular solution. This article describes the problem of finding a special solution of a differential Clairaut-type equation in partial derivatives with a special right-hand side. A special solution is found for a Clairaut-type equation, when the right-hand side has the form of a logarithmic function of the product of partial derivatives of the unknown function and their powers. Note that in both cases, for a given choice of the function of the derivatives in the equation, it is possible to solve the system of equation that determines the particular solution of the equation. The search for specific solutions for specific functions remains poorly understood and represents a promising direction for further research.