PDEMOC11

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(x+u)u_{x}+(y+u)u_{y}=0\,

The characteristics are {\frac  {dx}{dt}}=x+z,{\frac  {dy}{dt}}=y+z,{\frac  {dz}{dt}}=0\,.

\ln(x+z)=t+c_{1},\ln(y+z)=t+c_{2},z(t)=c_{3}\,

x+z=c_{4}e^{t},y+z=c_{5}e^{t}\,

y+z=c_{6}(x+z)\,

{\frac  {y+z}{x+z}}=c_{6}\,

So we have two functions that are LI and constant on the surface. Set one equal to an arbitrary function \phi \, of the other:

z=u(x,y,z)=\phi \left({\frac  {y+z}{x+z}}\right)\,


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