(i) State the local existence theorem for the first order quasi-linear partial differential equation

$$\sum_{j=1}^{n} a_{j}(x, u) \frac{\partial u}{\partial x_{j}}=b(x, u)$$

which is to be solved for a real-valued function with data specified on a hypersurface $S$. Include a definition of "non-characteristic" in your answer.

(ii) Consider the linear constant-coefficient case (that is, when all the functions $a_{1}, \ldots, a_{n}$ are real constants and $b(x, u)=c x+d$ for some $c=\left(c_{1}, \ldots, c_{n}\right)$ with $c_{1}, \ldots, c_{n}$ real and $d$ real) and with the hypersurface $S$ taken to be the hyperplane $\mathbf{x} \cdot \mathbf{n}=0$. Explain carefully the relevance of the non-characteristic condition in obtaining a solution via the method of characteristics.

(iii) Solve the equation

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

with initial data $u(0, y)=-y$ prescribed on $x=0$, for a real-valued function $u(x, y)$. Describe the domain on which your solution is $C^{1}$ and comment on this in relation to the theorem stated in (i).