Formulation of fundamental natural laws and technological problems in the form of rigorous mathematical models is given prevalently in terms of differential equations. Therefore, differential equations is a most important discipline in mathematical education. Numerous ad hoc methods for solving various particular types of differential equations have been developed within 300 years. They are summarized in voluminous catalogues containing e.g. about 400 types of second-order ordinary differential equations together with special methods of their solution.
Sophus Lie showed that the old methods of integration could be deduced simply by means of his theory. In particular, Lie reduced the classical 400 types of equations to 4 types only!
However, students, teachers and engineers still use ad hoc methods presented in traditional texts instead of dealing with Lie's four canonical equations. "Often the less there is to justify a traditional custom, the harder it is to get rid of it" (M. Twain).
ALGA develops courses based on Lie group analysis. We developed new mathematical programmes using Lie group methods in differential equations and mathematical modelling in nonlinear sciences. Introduction of these courses into curricula attracted more than 100 students instead of 10 at Moscow Institute of Physics and Technology, and 60 instead of 12 at Blekinge Institute of Technology.