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.