Applications of Lie groups to differential equations (How to use Lie Groups to solve differential equations)

Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.

It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.