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 fa...
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| Format: | Article |
| Language: | English |
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Samara State Technical University
2019-06-01
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| Series: | Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki |
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| Online Access: | https://journals.eco-vector.com/1991-8615/article/viewFile/20636/16883 |
| _version_ | 1811330141957128192 |
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| author | Liliya Leonidovna Ryskina |
| author_facet | Liliya Leonidovna Ryskina |
| author_sort | Liliya Leonidovna Ryskina |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T15:56:07Z |
| format | Article |
| id | doaj.art-3c8b09afdf5647ab9e23a891fdf49b50 |
| institution | Directory Open Access Journal |
| issn | 1991-8615 2310-7081 |
| language | English |
| last_indexed | 2024-04-13T15:56:07Z |
| publishDate | 2019-06-01 |
| publisher | Samara State Technical University |
| record_format | Article |
| series | Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki |
| spelling | doaj.art-3c8b09afdf5647ab9e23a891fdf49b502022-12-22T02:40:40ZengSamara State Technical UniversityVestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki1991-86152310-70812019-06-0123239440110.14498/vsgtu167918056On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functionsLiliya Leonidovna Ryskina0Tomsk State Pedagogical UniversityIn 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.https://journals.eco-vector.com/1991-8615/article/viewFile/20636/16883partial differential equationsclairaut-type equationssingular solutions |
| spellingShingle | Liliya Leonidovna Ryskina On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki partial differential equations clairaut-type equations singular solutions |
| title | On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions |
| title_full | On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions |
| title_fullStr | On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions |
| title_full_unstemmed | On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions |
| title_short | On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions |
| title_sort | on singular solutions of a multidimensional differential equation of clairaut type with power and exponential functions |
| topic | partial differential equations clairaut-type equations singular solutions |
| url | https://journals.eco-vector.com/1991-8615/article/viewFile/20636/16883 |
| work_keys_str_mv | AT liliyaleonidovnaryskina onsingularsolutionsofamultidimensionaldifferentialequationofclairauttypewithpowerandexponentialfunctions |