Linearization of second order ordinary differential equations via Cartan's equivalence method (Q1119791)

The problem of characterizing those second order differential ordinary differential equations \(y''=F(x,y,y')\) which are linearizable by a point transformation is solved using Cartan's equivalence method. The result is nicely summarized in the Theorem: The following (local) conditions are equivalent: (1) The equation has SL(3,R) as its infinitesimal symmetry group; (2) The equation is linearizable; (3) F is a cubic polynomial in \(y'\) given by \(F=A+3By'+3C(y')^ 2+D(y')^ 3\) with some technical conditions on A, B, C and D which must be satisfied.