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I am given a 1st order partial differential equation $y{\partial \psi\over\partial x}+x{\partial \psi\over\partial y}=0$ subjected to boundary condition $\psi(x,0)=\exp(-x^2)$. I have found that a solution is $\psi(x,y)=\exp(y^2-x^2)$. But I am asked when the solution is unique. Could someone please explain how to answer this? Thanks.

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    It would be useful if you could tell us how you came up with your solution. Perhaps by the method of characteristics?2011-12-18
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    Also, please include the domain of your PDE. Is it all of $\mathbb{R}^2$? The answer to your question will depend on the domain.2011-12-18

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