sin(3x)*y^3 is harmonic?

Question

I want to know if this function is harmonic or not

Answer 1

Let u=sin(3x)y³,

∂u/∂x=3cos(3x)y³,

∂u/∂y=3sin(3x)y²,

∂²u/∂x²=-9sin(3x)y³,

∂²u/∂y²=6sin(3x)y.

Laplace equation:

∇²u=Δu=∂²u/∂x²+∂²u/∂y²≡0 for harmonic functions.

In this case, ∂²u/∂x²+∂²u/∂y²=-9sin(3x)y³+6sin(3x)y=3ysin(3x)(2-3y²)≢0, so the given function is not harmonic.