PDEMOC3

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y^{{-1}}u_{x}+u_{y}=0,u(x,1)=x^{2}\,

Let x=x(y)\, so u=u(x(y),y)\, and let x(1)=x_{1}\,.


Now u_{y}=u_{x}x'(y)+u_{y}\,.

Let u_{y}=u_{x}x'(y)+u_{y}=y^{{-1}}u_{x}+u_{y}=0\, so u\, is constant with respect to y\, and u(x(y),y)=u(x(1),1)=x_{1}^{2}\,.


x'(y)=y^{{-1}}\,

x(y)=\ln y+x_{1}\,


The solution is:

u(x,y)=(x-\ln y)^{2}\,


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