Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 32E / Solution

Write $\xi =V_1$ and $\eta =\varphi$. The vector fieldgenerates a Lie point symmetry of an ordinary differential equation if its local flow maps solution graphs to solution graphs. Its $n$th prolongation of a vector field iswhereandis the total derivative operator. The infinitesimal invariance criterion is

Put $p=u^{\prime }$ and $q=u^{\prime \prime }$. Direct calculation givesandForthe invariance condition isafter imposing $q=p^2/u-u^2$. The coefficient of $p^3$ isIt must vanish identically, soSince $u>0$, integration givesfor functions $F,G$.

Now takeThenand substitution givesThus the infinitesimal invariance criterion holds.

The generated one-parameter transformation is, for $c\ne 0$,while for $c=0$ it is $\tilde{x}=x+d\epsilon$, $\tilde{u}=u$. Combining these transformations gives the two-parameter symmetry groupThis is the affine-scaling symmetry of u double prime equals u prime squared over u minus u squared.