Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 32C / Solution

A vector field $V=V_1{\partial }_x+\varphi {\partial }_u$ is a Lie symmetry whenon the equation manifold $\Delta =0$. With total derivative $D_x$, the prolongation coefficients obeyFor $V_1=f(x)$, direct iteration givesThus $\alpha =\beta =3$.

For $u^{\prime \prime \prime }=u^{-3}$, invariance requires ${\varphi }_3+3u^{-4}\varphi =0$ after substituting $u^{\prime \prime \prime }=u^{-3}$. Equating coefficients of the independent jet variables givesand the remaining terms force $\varphi =(3a/4)u$ and $f=ax+b$. Hence the Lie algebra is spanned by