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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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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 Clairaut-type 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 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.

Keywords: partial differential equations, Clairaut-type equations, singular solutions

DOI: https://doi.org/10.14498/vsgtu1679

Full text: PDF file (880 kB) (published under the terms of the Creative Commons Attribution 4.0 International License)
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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 Clairaut-type 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 Clairaut-type 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 394--401
\mathnet{http://mi.mathnet.ru/vsgtu1679}
\crossref{https://doi.org/10.14498/vsgtu1679}


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  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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