Charpit's equations ($\dot{x}=F_p,\dot{y}=F_q,\dot{u}=p{F}_p+q{F}_q$)

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: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 $p_0{x}_0^{\prime }+q_0{y}_0^{\prime }=u_0^{\prime }$. The non-characteristic condition is $x_0^{\prime }{F}_q-y_0^{\prime }{F}_p\ne 0$. It makes the projection onto $(x,y)$ locally invertible. Compatible initial contact data then propagate as $du=pdx+qdy$, reconstructing a classical solution. Multiple permitted initial normal derivatives can give different local branches; loss of the projection Jacobian determinant marks a characteristic caustic.

For initial data $u(x,0)=1+x^3/4$ on $0\le x\le 1$, the characteristic surface has $x=s{e}^r$, $y=3s^2(1-e^{-2r})/2$, $u=1+s^3(e^{-3r}-3e^{-r}/4)$. Its projection determinant is $3s^2{e}^r(2e^{-2r}-1)$, so the sheet connected to the initial data loses local invertibility at $r=\log 2/2$. The regular sheet has $r<\log 2/2$ and projects to $0<x<\sqrt{2}$, $3(1-x^{-2})/2<y<3x^2/8$. Its upper boundary is a fold point of a critical manifold caustic. Solving the inverse yields $s^2=[x^2+\sqrt{x^4-(8/3)x^{2}y}]/2$; the other square-root branch lies on the folded sheet and cannot simply be joined as a unique smooth solution. The initial point $s=0$ is already characteristic, illustrating the need to inspect endpoints as well as the interior of a Cauchy curve.