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.
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