2005 # A Lie Symmetry Connection between Jacobi's Modular Differential Equation and Schwarzian Differential Equation

**Abstract:** In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi's modular differential equation. A six-dimensional Lie symmetry algebra is obtained and its symmetry generators are found to be given in terms of elliptic integrals. As a consequence the transformation between Jacobi's modular differential equation and the well-known Schwarzian differential equ…

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“…9.4. Good examples of Lie's symmetries application to Chazy's equation (42) and many other Jacobi's 'modular/elliptic' equations are presented in nice works [25] and [68]. As with solutions (33) and (55) we can write solutions to system (54) that respect Jacobi's identity (43).…”

confidence: 99%

“…9.4. Good examples of Lie's symmetries application to Chazy's equation (42) and many other Jacobi's 'modular/elliptic' equations are presented in nice works [25] and [68]. As with solutions (33) and (55) we can write solutions to system (54) that respect Jacobi's identity (43).…”

confidence: 99%

“…Consequences. An interrelation between (74) and Jacobi's formula (68) needs to be understood if the elliptic curve (66) has already been given in the canonical form (67) as…”

confidence: 99%

“…As far as we know, this is only the second appearance of special functions in the generators of Lie symmetries admitted by a nonlinear differential equation. The preceding example, which involves elliptic integrals, can be found in [55].…”

confidence: 99%