Suppose $\psi^{s}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ is a smooth one-parameter group of transformations acting on $\mathbb{R}^{2}$, with infinitesimal generator

$$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$$

(a) Define the $n^{\text {th }}$prolongation $\operatorname{Pr}^{(n)} V$ of $V$, and show that

$$\operatorname{Pr}^{(n)} V=V+\sum_{i=1}^{n} \eta_{i} \frac{\partial}{\partial u^{(i)}}$$

where you should give an explicit formula to determine the $\eta_{i}$ recursively in terms of $\xi$ and $\eta$.

(b) Find the $n^{t h}$ prolongation of each of the following generators:

$$V_{1}=\frac{\partial}{\partial x}, \quad V_{2}=x \frac{\partial}{\partial x}, \quad V_{3}=x^{2} \frac{\partial}{\partial x}$$

(c) Given a smooth, real-valued, function $u=u(x)$, the Schwarzian derivative is defined by,

$$S=S[u]:=\frac{u_{x} u_{x x x}-\frac{3}{2} u_{x x}^{2}}{u_{x}^{2}}$$

Show that,

$$\operatorname{Pr}^{(3)} V_{i}(S)=c_{i} S,$$

for $i=1,2,3$ where $c_{i}$ are real functions which you should determine. What can you deduce about the symmetries of the equations: (i) $S[u]=0$, (ii) $S[u]=1$, (iii) $S[u]=\frac{1}{x^{2}}$ ?