Cauchy Data query. Let $$\ y \dfrac {\partial u}{\partial x} - \ x\dfrac {\partial u}{\partial y}=0.$$ The characteristic curve obtains the function $y^2+x^2=\eta^2 $, i.e., a circle of radius $\eta$ and centre $(0,0)$.
I have to explain why the Cauchy data $u(x,0)=f(x),\, x\geq 0$ provides a unique solution. The solution yielded is $u(x,y)=F(x^2+y^2)$. I then need to find another example of Cauchy data that gives a unique solution.
I then need to solve $$\ y \dfrac {\partial u}{\partial x} - \ x\dfrac {\partial u}{\partial y}=1.$$ I obtain $$u(x,y)=F(x^2+y^2)+\arctan(|y|/x),$$ but how can this show it is impossible to find a solution in ${\bf R}^2$?
Any help? Thanks :)


I have to explain why the Cauchy data $u(x,0)=f(x),\, x\geq 0$ provides a unique solution.

Because to get a solution at any point $(x,y)$ you need to go back along your characteristic starting at the point $(x,y)$ to the initial data. Now note that any point in ${\bf R}^2$ can be connected by a unique characteristic with the data on the half-line $x\geq 0$.

I then need to find another example of Cauchy data that gives a unique solution.

If point one is clear it should not be too difficult to come up with an example.

but how can this show it is impossible to find a solution in  ${\bf R}^2$?

maybe because your solution is not defined at $(0,0)$? 
(btw, should not it be $\arctan \frac{x}{y}$, and not $\arctan \frac{y}{x}$ as in your solution?)
