(a) Solve the equation

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

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

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

where $f$ is a smooth function. You should discuss the domain on which the solution is smooth. For which functions $f$ can the solution be extended to give a smooth solution on the upper half plane $\{y>0\}$ ?

(b) Solve the equation

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

together with the boundary condition on the unit circle:

$$u(x, y)=x \quad \text { when } \quad x^{2}+y^{2}=1$$