I am trying to solve this problem: $u_x+2xu_y$=y, with u(0,y) = 1+$y^2$ for $\frac{-1}{2} < y < \frac{1}{2}$ and determine the greatest region in which a unique solution exists.
I have found the solution to this problem as well as its characteristics s = x and t = y-$x^2$, but I have not idea on how to find the region in which the solution is unique. I know that I should find a region in which the characteristics intersect the initial data curve exactly once, but here the initial data curve i.e. y-axis happens to be one of the characteristics i.e. x = 0. If I remove this characteristic, then the initial data curve will also be removed, and (I think) the solution will not exist at all, let alone being unique.
Can anyone help me on this question? Any help is greatly appreciated!