
Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2019, Volume 23, Number 2, Pages 394–401
(Mi vsgtu1679)




Short Communication
On singular solutions of a multidimensional differential
equation of Clairauttype with power and exponential functions
L. L. Ryskina^{} ^{} Tomsk State Pedagogical University, Tomsk, 634061, Russian Federation
Abstract:
In the theory of ordinary differential equations, the Clairaut
equation is well known. This equation is a nonlinear 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
righthand side. The work is organized as follows. The introduction
provides a brief overview of some of the current results relating to
the study of Clairauttype equations in field theory and classical
mechanics. The first part provides general information about
differential equations of the Clairauttype 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
Clairauttype equations. The main result of the work is to find
singular solutions of equations containing power and exponential
functions.
Keywords:
partial differential equations, Clairauttype equations, singular solutions
DOI:
https://doi.org/10.14498/vsgtu1679
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Bibliographic databases:
UDC:
517.957
MSC: 35F20 Received: March 6, 2019 Revised: May 14, 2019 Accepted: June 10, 2019 First online: June 28, 2019
Citation:
L. L. Ryskina, “On singular solutions of a multidimensional differential
equation of Clairauttype with power and exponential functions”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:2 (2019), 394–401
Citation in format AMSBIB
\Bibitem{Rys19}
\by L.~L.~Ryskina
\paper On singular solutions of a multidimensional differential
equation of Clairauttype with power and exponential functions
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2019
\vol 23
\issue 2
\pages 394401
\mathnet{http://mi.mathnet.ru/vsgtu1679}
\crossref{https://doi.org/10.14498/vsgtu1679}
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