(a) State a local existence theorem for solving first order quasi-linear partial differential equations with data specified on a smooth hypersurface.

(b) Solve the equation

$$\frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=0$$

with boundary condition $u(x, 0)=f(x)$ where $f \in C^{1}(\mathbb{R})$, making clear the domain on which your solution is $C^{1}$. Comment on this domain with reference to the noncharacteristic condition for an initial hypersurface (including a definition of this concept).

(c) Solve the equation

$$u^{2} \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$$

with boundary condition $u(x, 0)=x$ and show that your solution is $C^{1}$ on some open set containing the initial hypersurface $y=0$. Comment on the significance of this, again with reference to the non-characteristic condition.