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## Main Question or Discussion Point

Consider the PDE [tex] xu_x + yu_y = 4u. [/tex] Suppose that we want to find the solution that satisfies [itex]u=1[/itex] on the circle [itex]x^2 + y^2 = 1[/itex] using the method of characteristics.

I have read that the boundary condition can be parameterized as

[tex] x=\sigma, \qquad y=(1-\sigma^2)^{1/2}, \qquad u=1.[/tex]

My question is this: couldn't we also have [itex]y=-(1-\sigma^2)^{1/2}[/itex]?? Don't we have to assume that it could be either the positive or negative square root ([itex] y = \pm\sqrt{1-x^2} [/itex])?

I have read that the boundary condition can be parameterized as

[tex] x=\sigma, \qquad y=(1-\sigma^2)^{1/2}, \qquad u=1.[/tex]

My question is this: couldn't we also have [itex]y=-(1-\sigma^2)^{1/2}[/itex]?? Don't we have to assume that it could be either the positive or negative square root ([itex] y = \pm\sqrt{1-x^2} [/itex])?