# Laplace u => Harmonic function

## Main Question or Discussion Point

In R2, I am to find all homog. polys (deg 2) that are harmonic.

the earlier homework included something like u = xy, show it's harmonic. EASY as pi. But I'm not really sure how to set this problem up. I understand the concept that a harmonic function will look like $$\nabla^{2} u = 0$$, but I'm not sure how to find all polys of degree 2.

I started out doing something like

$$u = a_{2}x^{2}_{1} + a_{1}x_{1} + a_{0} + b_{2}x^{2}_{2} + b_{1}x_{2} + b_{0}+...$$

and taking the partial with respect to each xi but that's not getting me very far.

Any suggestions?

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Ben Niehoff
$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F$$