Abstract
The following theorem is proved by investigating the Janet bases of determining systems: In order that a 3rd order quasilinear ordinary differential equation has a seven-parameter symmetry group it must have a certain structure, and a set of necessary and sufficient conditions for its coefficients must be satisfied. This theorem generalizes similar results for linear equations and for quasilinear equations of 2nd order. It is shown how this theorem facilitates the computation of closed form solutions.
Zusammenfassung
Es wird der folgende Satz bewiesen mit Hilfe der Eigenschaften von Janet Basen: Damit eine quasilineare gewöhnliche Differentialgleichung 3ter Ordnung eine sieben-parametrige Symmetriegruppe hat, muß sie eine bestimmte Skruktur haben und eine Menge notwendiger und hinreichender Bedingungen für die Koeffizienten muß erfüllt sein. Dieser Satz verallgemeinert ähnliche Ergebnisse für lineare Gleichungen und für quasilineare Gleichungen 2ter Ordnung. Es wird gezeigt, wie dieser Satz die Bestimmung geschlossener Lösungen erleichtert.
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Schwarz, F. On 3rd order ordinary differential equations with maximal symmetry group. Computing 57, 273–280 (1996). https://doi.org/10.1007/BF02247410
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DOI: https://doi.org/10.1007/BF02247410