A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries
In this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in terms of Lie symmetries. To get...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1312 |
| _version_ | 1797531002721009664 |
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| author | Daniele Ritelli |
| author_facet | Daniele Ritelli |
| author_sort | Daniele Ritelli |
| collection | DOAJ |
| description | In this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in terms of Lie symmetries. To get the complete solution, hypergeometric functions come into play, which, of course, were unknown in Riccati’s time. Re-discovering the method introduced by Riccati, called by himself dimidiata separazione (splitted separation), we arrive at the closed form integration of a differential equation, more general to the one treated in Riccati’s contribution, and which also does not appear in the known repertoires. |
| first_indexed | 2024-03-10T10:37:55Z |
| format | Article |
| id | doaj.art-4117f1bc95004b48a192492fe040bf0b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:37:55Z |
| publishDate | 2021-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-4117f1bc95004b48a192492fe040bf0b2023-11-21T23:10:25ZengMDPI AGMathematics2227-73902021-06-01911131210.3390/math9111312A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie SymmetriesDaniele Ritelli0Department of Statistics, University of Bologna, Via Belle Arti 41, 40126 Bologna, ItalyIn this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in terms of Lie symmetries. To get the complete solution, hypergeometric functions come into play, which, of course, were unknown in Riccati’s time. Re-discovering the method introduced by Riccati, called by himself dimidiata separazione (splitted separation), we arrive at the closed form integration of a differential equation, more general to the one treated in Riccati’s contribution, and which also does not appear in the known repertoires.https://www.mdpi.com/2227-7390/9/11/1312splitted separationLie symmetriesgauss hypergeometric functions |
| spellingShingle | Daniele Ritelli A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries Mathematics splitted separation Lie symmetries gauss hypergeometric functions |
| title | A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries |
| title_full | A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries |
| title_fullStr | A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries |
| title_full_unstemmed | A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries |
| title_short | A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries |
| title_sort | forgotten differential equation studied by jacopo riccati revisited in terms of lie symmetries |
| topic | splitted separation Lie symmetries gauss hypergeometric functions |
| url | https://www.mdpi.com/2227-7390/9/11/1312 |
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