I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential equations. I am aware that he refers to V.I. Arnold's "Geometrical Methods in the Theory of ODEs" for details, but couldn't find a satisfactory example.
I am looking for a concrete example where a contact geometric point of view helps in answering a question about an ODE.