Proposition

Suppose $u(x, y)$ is a function $\mathbb{R}^2 \rightarrow \mathbb{R}$ that depends only on $x$. When is $u$ harmonic?

Solution

Since $u$ does not depend on $y$, $u_{yy} = 0$. Therefore, $u_{xx}$ must be 0 if $u$ is harmonic. This is only satisfied by $u(x, y) = ax + b$ for some $a, b \in \mathbb{R}$.