But in this process, we met some difficulties, so by taking into account some assumptions we obtained the symmetries of ODEs which are in the special form, and also their solutions.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, the first-order ODEs which have no systematic way to find their Lie point symmetries -unlike higher order ODEs which have systematic ways- are reconsidered.
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Key Findings & Data:As a first step, we considered first order PDEs which correspond to these equations by introducing reduced characteristic Q that used in the Lie’s theory.
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Conclusion & Impact:But in this process, we met some difficulties, so by taking into account some assumptions we obtained the symmetries of ODEs which are in the special form, and also their solutions.
In this paper, the first-order ODEs which have no systematic way to find their Lie point symmetries -unlike higher order ODEs which have systematic ways- are reconsidered. As a first step, we considered first order PDEs which correspond to these equations by introducing reduced characteristic Q that used in the Lie’s theory. Following this step, we tried to obtain solutions of the PDEs using their Lie point symmetries. But in this process, we met some difficulties, so by taking into account some assumptions we obtained the symmetries of ODEs which are in the special form, and also their solutions.
Açil, M., Konuralp, A., Bildik, N. (2017). Finding The Lie Symmetries of Some First-Order Odes Via Induced Characteristic. Celal Bayar Üniversitesi Fen Bilimleri Dergisi, Cilt: 13 Sayı: 2, 275–278. https://doi.org/10.18466/cbayarfbe.319747
Bildik N.; Açil M. On the Lie Symmetries of First-order Ordinary Differential Equations. AIP Proceeding. 2013; 1558, 2575–2578.
Hydon P.E. Symmetry Methods for Differential Equations (A Beginner’s Guide). Cambridge Texts in Apllied Mathematics, 2000.
Cheb-Terrab E.S.; Kolokolnikov T. First-order Ordi-nary Differential Equations. Symmetries and Linear Transformation. European Journal of Applied Mathe-matics. 2003; 14:2 231-246.
Cheb-Terrab E.S.; Roche A.D. Symmetries and first-order ODE Patterns, Computer Physics Communica-tions. 1998; 113:239-260.
Andrei D. Polyanin, Valentine F. Zaitsev,Hand Book of Exact Solutions for Ordinary Differential Equations (Second Edition), Chapman - Hall/CRC, 2002.