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Method of characteristics

Hello :-)

I don't quite understand when there is a unique solution, no solution or infinite solutions (for aUx + bUy=c)

I tried to do this excercise (see the attached file): (btw, the domain is x^2+y^2>=1)

I found the characteristic curves and the inital value curve (a circle of radius 1).

As I understand for a unique solution to exist, a charactersitc curve must intersect with the initial value curve exactly once.

So I drew both of the curves and according to the drawing only characteristic curves for which c=1 or c=-1 obey this condition. Does this mean a unique solution exist only on them?

What about the other characteristic curves (c greater than 1), which do not intersect the initial value curve at all, does this mean no solution exists of them?

So is it safe to say a unique solution exists only on the characteristic curvers for which c=1 or c=-1?

I also checked the transervasality condition (did I do it correctly?). I got that J isn't 0 which mean the transervasality condition holds. But what information does that give me?

What about r=0 , r=pie/2, r=pie , it seems that J isn't defined for those values.. what does that mean?

Also, I was asked to find a second boundary condition so that the solution would be defined for all x,y in the domain. How do I do that?

Thank you in advance !