Let $u(x,y)=x^{3}+ax^{2}y+bxy^{2}+2y^{3}$ be a harmonic function and $v(x,y)$ be its harmonic conjugate. If $v(0,0)= 1$, then find $|a+b+v(1,1)|$.
For finding $a$ and $b$
Let $u(x,y)=x^{3}+ax^{2}y+bxy^{2}+2y^{3}$ be a harmonic function , then it satisfy Laplacian equation.
From that I got,
$6x+12y+2ay+2bx=0x+0y\tag{1}.$
Can I equate the coefficients and find $a$ and $b$? Is there any logical error?
Rest of the solution, I can do.