In Lie Gourp wikipedia entry (https://en.wikipedia.org/wiki/Lie_group) it is said:

The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject.

But did Lie's theory finally manage to unify the entire field of ordinary differential equations? Or maybe some version of it?...

  • $\begingroup$ No unification. But it is helpful in finding analytic solutions to ODE and PDE, and is used in the theory of integrable systems. The modern term is Lie group analysis of DE, see Ibragimov's book. $\endgroup$ – Conifold Mar 30 at 6:20
  • $\begingroup$ @Conifold even no variants of Lie groups/Lie theory can do it? $\endgroup$ – Forsete Apr 1 at 18:58
  • $\begingroup$ Do what, unification? No, most DE do not have many symmetries, if any at all. $\endgroup$ – Conifold Apr 1 at 19:39

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