Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 33A / d

Put $q=u/x^2$, so $F=xG(q)$ andSubstitution into the Lie-symmetry determining equation for a first-order ordinary differential equation and separation with respect to the arbitrary function $G$ givesThese equations imply $\xi =Cx$ and $\eta =2Cu$. Apart from the zero generator, the common symmetry for arbitrary $G$ is therefore generated byIts one-parameter group is the scaling symmetrywhich leaves $u/x^2$ invariant. Special choices of $G$ can have additional symmetries, but this is the complete symmetry valid for an arbitrary smooth $G$.