First-order partial differential equation ($F(x,u,Du)=0$)

A partial differential equation whose highest partial derivatives have order one. For $a(x,y)u_x+b(x,y)u_y=c(x,y,u)$, the method of characteristics follows the ordinary differential equations $\dot{x}=a$, $\dot{y}=b$, $\dot{u}=c$.