ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONS
Background. There is no problem in finding a general solution to the ordinary differential Clairaut equation. The corresponding procedure is described in details in the theory of ordinary differential equations. Except of general solution being the family of linear functions, special (singular) so...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Penza State University Publishing House
2020-06-01
|
| Series: | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
| Subjects: |
| _version_ | 1818772567781015552 |
|---|---|
| author | L. A. Zhidova |
| author_facet | L. A. Zhidova |
| author_sort | L. A. Zhidova |
| collection | DOAJ |
| description | Background. There is no problem in finding a general solution to the ordinary
differential Clairaut equation. The corresponding procedure is described in details in the theory of ordinary differential equations. Except of general solution being the
family of linear functions, special (singular) solutions for the ordinary differential
Clairaut equation may exist for which are no general methods to find them. This is
evidenced by a very meager list in the accessible scientific literature of the types of
Clairaut equations for which special solutions can be explicitly constructed. Therefore,
it seems as an actual task finding and studies special solutions to the Clairaut
equations. Goal of the present paper is finding and studies of special solutions to the
Clairaut equations in the theory of ordinary differential equations and of partial differential
equations and setting relations among special solutions to the Clairaut
equation in the theory of ordinary differential equations and of partial differential
equations.
Results. We discuss basic concepts and solutions to the Clairaut equations in the
theory of ordinary differential equations and of partial differential equations. We
find relations between singular solutions to the Clairaut equations in the theory of
ordinary differential equations and of partial differential equations.
Conclusions. It is proved that there exists the relation between singular solutions
to the Clairaut equations in the theory of ordinary differential equations and
of partial differential equations in the class of special dependences of right hand
side of the Clairaut in the theory of partial differential equations. The number
of such relations is defined by the number of known special solutions to the
Clairaut equations in the theory of ordinary differential equations and all their
possible combinations. |
| first_indexed | 2024-12-18T10:11:25Z |
| format | Article |
| id | doaj.art-ef9b04ee28aa40df9deabf9c092d2044 |
| institution | Directory Open Access Journal |
| issn | 2072-3040 |
| language | English |
| last_indexed | 2024-12-18T10:11:25Z |
| publishDate | 2020-06-01 |
| publisher | Penza State University Publishing House |
| record_format | Article |
| series | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
| spelling | doaj.art-ef9b04ee28aa40df9deabf9c092d20442022-12-21T21:11:25ZengPenza State University Publishing HouseИзвестия высших учебных заведений. Поволжский регион: Физико-математические науки2072-30402020-06-01210.21685/2072-3040-2020-2-4ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONSL. A. Zhidova0Tomsk State Pedagogical UniversityBackground. There is no problem in finding a general solution to the ordinary differential Clairaut equation. The corresponding procedure is described in details in the theory of ordinary differential equations. Except of general solution being the family of linear functions, special (singular) solutions for the ordinary differential Clairaut equation may exist for which are no general methods to find them. This is evidenced by a very meager list in the accessible scientific literature of the types of Clairaut equations for which special solutions can be explicitly constructed. Therefore, it seems as an actual task finding and studies special solutions to the Clairaut equations. Goal of the present paper is finding and studies of special solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations and setting relations among special solutions to the Clairaut equation in the theory of ordinary differential equations and of partial differential equations. Results. We discuss basic concepts and solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations. We find relations between singular solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations. Conclusions. It is proved that there exists the relation between singular solutions to the Clairaut equations in the theory of ordinary differential equations and of partial differential equations in the class of special dependences of right hand side of the Clairaut in the theory of partial differential equations. The number of such relations is defined by the number of known special solutions to the Clairaut equations in the theory of ordinary differential equations and all their possible combinations.ordinary differential equationspartial derivative equationsclairaut equationssingular solutions |
| spellingShingle | L. A. Zhidova ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONS Известия высших учебных заведений. Поволжский регион: Физико-математические науки ordinary differential equations partial derivative equations clairaut equations singular solutions |
| title | ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONS |
| title_full | ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONS |
| title_fullStr | ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONS |
| title_full_unstemmed | ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONS |
| title_short | ON SINGULAR SOLUTIONS OF CLAIRAUT EQUATIONS IN THE THEORY OF ORDINARY DIFFERENTIAL AND PARTIAL DERIVATIVE EQUATIONS |
| title_sort | on singular solutions of clairaut equations in the theory of ordinary differential and partial derivative equations |
| topic | ordinary differential equations partial derivative equations clairaut equations singular solutions |
| work_keys_str_mv | AT lazhidova onsingularsolutionsofclairautequationsinthetheoryofordinarydifferentialandpartialderivativeequations |