Verify that the function $u(x,y)=x^2-y^2-y$ is harmonic

Question

How do I verify that the function $$u(x,y)=x^2-y^2-y$$ is harmonic

Answer 1

$$u_{xx}=2\;\;,\;u_{yy}=-2\Longrightarrow \Delta u=...?$$

Answer 2

Let $u(x,y)=ax^2+2hxy+by^2+2gx+2fy+c=0$

So, $u_x=2ax+2hy+2g, u_{xx}=2a$

Similarly, $u_{yy}=2b$

Using this, we need $u_{xx}+u_{yy}=0$ and $u_{xx}+u_{yy}=2(a+b)\implies b=-a$

Answer 3

It is the real part of $z^2 +iz$, therefore harmonic.