BSU bulletin. Mathematics, Informatics
, A CLAIRAUT-TYPE DIFFERENTIAL EQUATION IN PARTIAL DERIVATIVES WITH A POWER FUNCTION // BSU bulletin. Mathematics, Informatics. - 2019. №1. . - С. 41-48.
A CLAIRAUT-TYPE DIFFERENTIAL EQUATION IN PARTIAL DERIVATIVES WITH A POWER FUNCTION
The article is aimed at the study of Clairaut-type differential equations in partial deriva- tives of the first order with a special right-hand side. Finding a general solution for an ordinary Clairaut differential equation is not difficult, and it is described in detail in courses on the theory of ordinary differential equations. In addition to the general solu- tion, which involves a family of integral lines, there is a special solution for an ordinary Clairaut differential equation, represented by the envelope of this family. Within the framework of the theory of differential equations in partial derivatives there are Clair- aut-type differential equations, which are multidimensional generalizations of an ordi- nary Clairaut differential equation. Note that for a Clairaut-type partial differential equa- tion there is no always a special solution. The article is devoted to the problem of the de- scription of a particular solution for Clairaut-type equations, which right side has the form of a power function of the product of n factors.
partial differential equations; Clairaut-type differential equations; singu- lar solutions; power function.
List of references:
Zaitsev V. F., Polyanin A. D. Spravochnik po obyknovennym differentsial'nym uravneniyam [Reference Book on Ordinary Differential Equations]. Moscow: Fizmatlit Publ., 2002. 256 p.
Stepanov V. V. Kurs differentsialnykh uravnenii [A Course in Differential Eq- uations]. Moscow: Fizmatlit Publ., 1965. 512 p.
Elsgolts L. E. Differentsialnye uravneniya i variatsionnoe ischislenie [Differ- ential Equations and Variational Calculus]. Moscow: Nauka Publ., 1969. 424 p.
Zaitsev V. F., Polyanin A. D. Spravochnik po differentsialnym uravneniyam v chastnykh proizvodnykh pervogo poryadka [Reference Book on Differential Equations in Partial Derivatives of the First Order]. Moscow: Fizmatlit Publ., 2003. 416 p.
Kamke E. Spravochnik po differentsialnym uravneniyam v chastnykh proiz- vodnykh pervogo poryadka [Reference Book on Differential Equations in First-Order Partial Derivatives]. Moscow: Nauka Publ., 1966. 260 p. (transl. from German)
Courant R., and Hilbert D. Methods of Mathematical Physics. V. 2. Partial Differential Equations. New York: Interscience,1962. 830 p.
Lavrov P. M., Merzlikin B. S. Legendre Transformations and Clairaut-Type Equations. Physics Letters. 2016. V. 756. P. 188–193.
Lavrov P. M., Merzlikin B. S. Loop Expansion of the Average Effective Ac- tion in the Functional Renormalization Group Approach. Phys. Rev. 2015. V. 92. No. 8. .
Zhidova L. A., Zyryanova O. V., Kholmukhammad F. Differentsialnye urav- neniya v professionalnoi podgotovke uchitelya matematiki [Differential Equations in the Professional Training of Mathematics Teachers]. TSPU Bulletin. 2017. No. 1 (178). Pp. 75–78.
Rakhmelevich I. V. O resheniyakh mnogomernogo uravneniya Klero s mul- tiodnorodnoi funktsiei ot proizvodnykh [On the Solutions of Multi-Dimensional Clair- aut Equation with a Multi-Homogeneous Function of Derivatives]. Izvestiya Sara- tovskogo universiteta. Novaya seriya. Ser. Matematika, mekhanika, informatika. 2014. V. 14. No. 4–1. Pp. 374–381.