(a) Solve the equation, for a function $u(x, y)$,

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

together with the boundary condition on the $x$-axis:

$$u(x, 0)=x$$

Find for which real numbers $a$ it is possible to solve $(*)$ with the following boundary condition specified on the line $y=a x$ :

$$u(x, a x)=x$$

Explain your answer in terms of the notion of characteristic hypersurface, which should be defined.

(b) Solve the equation

$$\frac{\partial u}{\partial x}+(1+u) \frac{\partial u}{\partial y}=0$$

with the boundary condition on the $x$-axis

$$u(x, 0)=x$$

in the domain $\mathcal{D}=\left\{(x, y): 0<y<(x+1)^{2} / 4,-1<x<\infty\right\}$. Sketch the characteristics.