ABSTRACT
After discussing the various integration methods for ode’s developed by Lie it appears appropriate to conclude with a few words of comparison, assess its advantages and limitations, and also put them into proper perspective. Furthermore, several directions of future research are outlined. Comparison of Lie’s Methods. There are basically two alternative methods for solving differential equations using its symmetries. Either an equation with a certain nontrivial symmetry is transformed into a canonical form the solution of which may be determined. Or the infinitesimal symmetry generators are determined explicitly, and are used to determine a set of first integrals from which the solution may be constructed. The relation between these two alternatives is most easily understood from the following drawing. The order n of the given quasilinear ode is two or three.
