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 $$ \begin{equation} \tag{6} \frac{\partial u}{\partial t} +\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y }=0, \end{equation} $$