Symmetries of second- and third-order ordinary differential equations

In order to apply Lie's symmetry theory for solving a differential equation it must be possible to identify the group of symmetries leaving the equation invariant. The answer is obtained in two steps. At first a classification of the possible symmetries of equations of the respective order is determined. Secondly a decision procedure is provided which allows to identify the symmetry type within this classification. For second-order equations the answer has been obtained by Lie himself. In this article the complete answer for quasilinear equations of order three is given. An important tool is the Janet base representation for the determining system of the symmetries.