I'm doing an exercise with no solution, the question says $u_x+xu_y=0$ for $x,y\in \mathbb{R}$, with initial value $u(x,0)=f(x)$, where $f$ is a real function.
Now the question ask me to impose some condition(s) to $f$ such that the solution exists. My idea is to first compute the characteristics along the $x$-axis, so $y=x^2/2+\zeta$ and thus if the solution exists, we must have $u(x,y)=f(\pm\sqrt{x^2-2y}).$ And I impose the condition that $f$ is an even function so the solution isn't multi-valued , as well as $f\in C^1(\mathbb{R}).$
Another question is to find the region of the $(x,y)$-plane such that the solution is uniquely determined by the boundary condition. And I think it should be the half space $x>0$ or $x<0,$ thus when we are parametrising along the $x$-axis ($y=0$), the equation $x^2/2+\zeta=0$ has only one root. (I'm not sure if this is correct, or do I need to specify further that the valid region is $x>0, y<0$ and $x<0, y<0.$) Can anyone help me with these 2 Qs?