Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$

Question

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$

I have concluded that $xu_x = yu_y$. Not sure how to proceed from here. Thanks for any help.

Answer 1

Hint: next show that $u$ is (locally) a function of $xy$, i.e. is constant on the curves $xy = c$.