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

(a) Define the generator of the transformation,

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

where you should specify $\xi$ and $\eta$ in terms of $\psi^{s}$.

(b) Define the $n^{\text {th }}$prolongation of $V, \operatorname{Pr}^{(n)} V$ and explicitly compute $\operatorname{Pr}^{(1)} V$ in terms of $\xi, \eta$.

Recall that if $\psi^{s}$ is a Lie point symmetry of the ordinary differential equation:

$$\Delta\left(x, u, \frac{d u}{d x}, \ldots, \frac{d^{n} u}{d x^{n}}\right)=0$$

then it follows that $\operatorname{Pr}^{(n)} V[\Delta]=0$ whenever $\Delta=0$.

(c) Consider the ordinary differential equation:

$$\frac{d u}{d x}=F(x, u),$$

for $F$ a smooth function. Show that if $V$ generates a Lie point symmetry of this equation, then:

$$0=\eta_{x}+\left(\eta_{u}-\xi_{x}-F \xi_{u}\right) F-\xi F_{x}-\eta F_{u}$$

(d) Find all the Lie point symmetries of the equation:

$$\frac{d u}{d x}=x G\left(\frac{u}{x^{2}}\right)$$

where $G$ is an arbitrary smooth function.