Abstract:
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.
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