Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 29D / b

A Lie point symmetry is a one-parameter transformation of $(x,u)$ which maps the graph of a function solving the differential equation to another such graph of a function, wherever the transformed independent variable is invertible along the graph of a function. Its induced transformation on derivatives is the prolongation. Infinitesimally the criterion is ${\mathrm{pr}}^{(n)}V(\Delta )=0$ on $\Delta =0$. The derivative coefficients can be computed recursively from ${\eta }^{(0)}=\eta$ by ${\eta }^{(j)}=D_x{\eta }^{(j-1)}-u^{(j)}{D}_x\xi$. This expresses the requirement that the transformed derivatives still satisfy the equation.