<s>[INST] Solutions to the Laplace Equation $\Delta u =0$, where $u= \log p$ Find all real solutions to the two dimensional Laplace equation $U_{xx} + U_{yy} =0$ of the form $u=\log p(x,y)$, where $p$ is a quadratic polynomial. 
Solution:
Let $p(x,y) = Ax^2 + By^2 +Cxy + D$ be a quadratic polynomial such that $A, B \not= 0$. Then 
$$U_{x} = \frac{2Ax + Cy}{\ln(10)(Ax^2 + By^2 +Cxy + D},$$ 
$$U_{xx} = \frac{2A ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2Ax + Cy)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2},$$
$$U_{y} =  \frac{2By + Cx}{\ln(10)(Ax^2 + By^2 +Cxy + D)},$$
$$U_{yy} = \frac{2B \ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2By + Cx)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2}.$$ 
This implies 
$$U_{xx} + U_{yy} = \frac{2A \ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2Ax + Cy)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2} + \frac{2B \ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2By + Cx)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2} = 0. $$ 
I feel like I am not doing this right. Is there a simpler way? Thanks. And also how do I find such solutions.
 [/INST] This is an heavy method, but you can continue.
$$U_{xx} + U_{yy} = \frac{2A ln(10)(Ax^2 + By^2 +Cxy + D) - ln(10)(2Ax + Cy)^2}{ln(10) (Ax^2 + By^2 +Cxy + D)^2} + \frac{2B ln(10)(Ax^2 + By^2 +Cxy + D) - ln(10)(2By + Cx)^2}{ln(10) (Ax^2 + By^2 +Cxy + D)^2} = 0$$
After simplification :
$$2A(Ax^2 + By^2 +Cxy + D)-(2Ax + Cy)^2 + 2B(Ax^2 + By^2 +Cxy + D)-(2By + Cx)^2=0$$
It is easy to see that $A=B=1\:;\:C=D=0$ is solution. Hense : $p(x,y)=x^2+y^2$
$$U(x,y)=log(x^2+y^2)$$
A simpler method consists in solving first the PDE :
$$U(x,y)=f(x+iy)+g(x-iy)$$
any functions $f$ and $g$.
In case of $f=g=log$ then $U=log(x+iy)+log(x-iy)$
$$U(x,y)=log(x^2+y^2)$$
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