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!