= Solution
Use the initial-curve parameter $s$ and a characteristic parameter $t$. <Charpit's equations> are
$$
\boxed{\dot x=F_p,\quad\dot y=F_q,\quad\dot u=pF_p+qF_q,\quad\dot p=-F_x-pF_u,\quad\dot q=-F_y-qF_u.}
$$
At $t=0$ prescribe $x=x_0(s)$, $y=y_0(s)$ and $u=u_0(s)$. The two initial <derivatives> $p_0,q_0$ must satisfy
$$
\boxed{F(x_0,y_0,u_0,p_0,q_0)=0,\qquad p_0x_0'+q_0y_0'=u_0'.}
$$
The second equation is the <chain rule> along the initial curve. For the usual local Cauchy construction also require the <non-characteristic> condition $x_0'F_q-y_0'F_p\ne0$. It ensures both local determination of an allowed branch of $p_0,q_0$ and local inversion of the projected characteristic coordinates.
Along a characteristic, the <chain rule> and the five equations give
$$
\frac{dF}{dt}=F_xF_p+F_yF_q+F_u(pF_p+qF_q)-F_p(F_x+pF_u)-F_q(F_y+qF_u)=0.
$$
Since $F=0$ initially, $\boxed{F\equiv0}$ along each constructed curve. With the contact compatibility $du=p\,dx+q\,dy$ propagated by this system, an invertible characteristic chart reconstructs $p=u_x,q=u_y$ and hence a local <classical solution>.
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