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?
$A=?$ $B=?$
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?
$A=?$ $B=?$
(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 $F(x,y) = -5x^5 + 50x^3y^2 - 25xy^4.$