{c}
{title2=$\dot x=F_p,\ \dot y=F_q,\ \dot u=pF_p+qF_q$}

For a first-order nonlinear <partial differential equation> $F(x,y,u,p,q)=0$, with $p=u_x,q=u_y$, Charpit's equations extend the <method of characteristics> to the five-dimensional space of position, value and gradient:
$$
\dot x=F_p,\quad\dot y=F_q,\quad\dot u=pF_p+qF_q,\quad\dot p=-F_x-pF_u,\quad\dot q=-F_y-qF_u.
$$
The chain rule gives $dF/dt=0$ by direct cancellation, so a trajectory starting on $F=0$ remains there. Initial data $u_0(s)$ on $(x_0(s),y_0(s))$ require both $F(x_0,y_0,u_0,p_0,q_0)=0$ and $p_0x_0'+q_0y_0'=u_0'$. The <non-characteristic> condition is $x_0'F_q-y_0'F_p\ne0$. It makes the projection onto $(x,y)$ locally invertible. Compatible initial contact data then propagate as $du=p\,dx+q\,dy$, reconstructing a classical solution. Multiple permitted initial normal derivatives can give different local branches; loss of the projection <Jacobian determinant> marks a <characteristic caustic>.


 Charpit's equations (source code)