Famous PDEs, Helmholtz' equation
Other famous partial differential equations are the Helmholtz (or eigenvalue) equation, here specialized to two
dimensions only
\begin{equation}
\tag{5}
-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}=\lambda u,
\end{equation}
the linear transport equation (in
2+1 dimensions) familiar from Brownian motion as well
\begin{equation}
\tag{6}
\frac{\partial u}{\partial t} +\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y }=0,
\end{equation}