nonlinear partial-differential-equations I have a problem with  partial differential equation
$$u_xu_y=xy$$
where $$u(0,y)=-y.$$
Therefore $x_0(s)=0$,$y_0(s)=s$,$u_0(s)=-s.$
I wrote my equation as $F(x,y,u,p,q)=pq-xy=0$ and defined the characteristic equations:
$$\frac{dx}{dt}=q$$
$$\frac{dy}{dt}=p$$
$$\frac{du}{dt}=2pq$$
$$\frac{dp}{dt}=y$$
$$\frac{dq}{dt}=x.$$
I know that
$F(x_0(s),y_0(s),u_0(s),p_0(s),q_0(s))=0$ and $p_0(s)x_0'(s)+q_0(s)y_0'(s)=u_0'(s)$,
then
$p_0(s)q_0(s)=0$ and $q_0(s)=-1$, therefore $p_0(s)=0.$ Then $p(t,s)=0$ and $q(t,s)=-1$ and then I solved characteristic equations: $x(t,s)=-t, y(t,s)=s, u(t,s)=-s.$ But u(x,y)=-y is not a solution, and I don't know what I did wrong.
 A: That's a good start, but the point where you go wrong is when you claim that $p(t,s)$ and $q(t,s)$ are constant functions; that's not correct, since their derivatives are $dp/dt=y$ and $dq/dt=x$, not $dp/dt=0$ and $dq/dt=0$.
To solve the characteristic equations correctly, note that the system decouples into two subsystems that you can solve separately,
$$
\frac{dx}{dt} = q
,\qquad
\frac{dq}{dt} = x
$$
and
$$
\frac{dy}{dt} = p
,\qquad
\frac{dp}{dt} = y
,
$$
together with the last equation for $u(t,s)$, which you can integrate rather easily once you have computed $q(t,s)$ and $p(t,s)$:
$$
\frac{du}{dt} = 2pq
.
$$
And of course you also need to use the right initial conditions at $t=0$ for all these equations, but you have already determined those conditions correctly ($x(0,s)=x_0(s)=0$, etc.), so I'm not writing them here.
There are many ways to solve the subsystems. For example, note that $d^2 x/dt^2=x$. Or use $x+q$ and $x-q$ as new variables.
Do you think you can take it from here?
A: I solved it and now I have:
$$x(t,s)=-\frac{1}{2}(e^t-e^{-t})$$
$$q(t,s)=-\frac{1}{2}(e^t+e^{-t})$$
$$y(t,s)=\frac{s}{2}(e^t+e^{-t})$$
$$p(t,s)=\frac{s}{2}(e^t-e^{-t})$$
and $u(t,s)=-\frac{s}{4}(e^{2t}+e^{-2t})-\frac{s}{2}.$
I checked the initial conditions and I think that it is a good solution, but I saw that
$$u(x,y)=xy-\frac{s}{2}$$ and I have a problem with $s$, because the function should depend only on $x$ and $y$.
