Let v be the harmonic conjugate of u=x³-3xy²-5y, then:
∂u/∂x=∂v/∂y=3x²-3y² (1)
∂u/∂y=-∂v/∂x=-6xy-5, so ∂v/∂x=6xy+5 (2).
Now we can partially integrate (1) and (2):
From (1): v=3x²y-y³+g(x).
From (2): v=3x²y+5x+h(y).
Therefore, v=3x²y-y³+5x, when g(x)=5x and h(y)=-y³.