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Find all solutions to $u_{xx}+u_{yy}=0$, where $u=\log p(x,y)$, with $p(x,y)$ a quadratic polynomial.

Assume $p(x,y)=ax^2+bxy+cy^2+dx+ey+f$, I computed $u_{xx}+u_{yy}$, then all coefficients in it have to be zero. But I failed to solve the correspnding system of algebraic equations...

I guess there would be a trickier way to find the solutions. Looking forward to your suggestions. Thanks!

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    (I had a wrong answer below, there were a bunch of comments on it which may be orphans now.)2012-09-08
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    @countinghaus Thank you, though. :)2012-09-08

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