ABSTRACT
Sections 7.1 and 7.2 provide a relatively complete outline of the theory, in part, because of its greater complexity as compared to the theory in the previous two chapters. These sections cover the definition and classification of two-parameter groups, invariance, and canonical coordinates. We shall find that every second-order ODE invariant under a two-parameter group can be fully solved in terms of quadratures. This is achieved by determining the classification, or type, for the group and introducing appropriate canonical coordinates that result in a separable form for the ODE which is twice integrable.
