ABSTRACT
The group classification of ordinary differential equations is based upon the enumeration of all possible Lie algebras of operators acting on the plane (x, y ). The basis operators of each algebra are simplified by a suitable change of variables. Algebras connected by a change of variables are called similar. Equations that admit similar algebras are also similar (equivalent) in the sense that they can be transformed into one another by a change of variables. The classification happens to be of an especially simple form in the case of second-order equations. In this case the dimension of a maximal admitted algebra has only the values 1, 2, 3, and 8. This chapter summarizes results for second-order equations admitting a three-dimensional algebra.
