Consider the coordinate transformation

$$g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})=(x \cos \epsilon-u \sin \epsilon, x \sin \epsilon+u \cos \epsilon)$$

Show that $g^{\epsilon}$ defines a one-parameter group of transformations. Define what is meant by the generator $V$ of a one-parameter group of transformations and compute it for the above case.

Now suppose $u=u(x)$. Explain what is meant by the first prolongation $\mathrm{pr}^{(1)} g^{\epsilon}$ of $g^{\epsilon}$. Compute $\mathrm{pr}^{(1)} g^{\epsilon}$ in this case and deduce that

$$\mathrm{pr}^{(1)} V=V+\left(1+u_{x}^{2}\right) \frac{\partial}{\partial u_{x}}$$

Similarly find $\mathrm{pr}^{(2)} V$.

Define what is meant by a Lie point symmetry of the first-order differential equation $\Delta\left[x, u, u_{x}\right]=0$. Describe this condition in terms of the vector field that generates the Lie point symmetry. Consider the case

$$\Delta\left[x, u, u_{x}\right] \equiv u_{x}-\frac{u+x f\left(x^{2}+u^{2}\right)}{x-u f\left(x^{2}+u^{2}\right)}$$

where $f$ is an arbitrary smooth function of one variable. Using $(\star)$, show that $g^{\epsilon}$ generates a Lie point symmetry of the corresponding differential equation.