Question

For what values of A and B is the polynomial \[f(x,y)= Ax ^{4}+6x ^{2}y ^{2}+By ^{4}\] function harmonic in the whole xy-plane?

Answer 1

\[df/dx = 4Ax ^{3}+12xy ^{2}\]
\[d^{2}f /dx ^{2}=12Ax ^{2}+12y ^{2}\]
\[df/dy=12x ^{2}y+4By ^{3}\]
\[d ^{2}f/dx ^{2}=12x ^{2}+12By ^{2}\]
the condition for harmonic function is:
\[d ^{2}f/dx ^{2}+d ^{2}f/dy ^{2}=0\]
so A=-1 and B=-1

Answer 2

@Wislar

Answer 3

So...
\[12Ax ^{2}+12y ^{2}+12x ^{2}+12By ^{2}=0\] \[12A+12=0 \\ 12B+12=0\]
Right?

Answer 4

right

Answer 5

Thanks!