Taking
xux+yuy=0 with u(x,y)=x on x2+y2=1
Due to the contour conditions we will consider the change of variables
x=rcosθy=rsinθ
with
dx=drcosθ−rsinθdθdy=drsinθ+rcosθdθ
we have
ux=urdxdr+uθdxdθuy=urdydr+uθdydθ
Here from the characteristic curves
xdx=ydy⇒dxdy=xy=tanθ
So, we obtain
xux+yuy=0⟺2rur=0⇒u(r,θ)=C+Φ(θ)
Note that
dxdrdydrdxdθdydθ=cosθ1=sinθ1=0=0
Now with the boundary conditions
u(1,θ)=cosθ⇒Φ(θ)=cosθ and C=0
and finally
u(r,θ)=cosθ
or in (x, y) coordinates, u(x,y)=x2+y2x
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