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I can't figure out this question:

For what values of the constants A and B is the polynomial function $F(x,y) = (-5)x^5 + Ax^{3}y^{2} + Bxy^{4}$ harmonic in the whole $xy$-plane?

$A=?$ $B=?$

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So, where did the computations of the Laplacian led you? –  Davide Giraudo Dec 9 '12 at 20:16
    
Do you know the definition of a harmonic function? Do you know how to calculate partial derivatives? –  Old John Dec 9 '12 at 20:25

1 Answer 1

(Just to save this question from going unanswered for ever)

For $F$ to be harmonic we need $$\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} = 0$$ throughout the plane.

Since $\frac{\partial^2 F}{\partial x^2} = -100x^3 + 6Axy^2$ and $\frac{\partial^2 F}{\partial y^2} = 2Ax^3 + 12Bxy^2$, this means that we need the function $-100x^3 + 6Axy^2 + 2Ax^3 + 12Bxy^2$ to be identically zero throughout the plane.

This implies that $2A-100=0$ and $12B+6B=0$, giving $a=50$ and $B=-25$, and thus the harmonic function is $F(x,y) = -5x^5 + 50x^3y^2 - 25xy^4.$

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