Solution
= Solution
Characteristics satisfy $\dot x=a$, $\dot y=b$, $\dot u=c$, with data prescribed on a transverse curve. If $(a,b)=(H_y,-H_x)$ then $dH/dt=H_xH_y-H_yH_x=0$. For the displayed field the divergence is zero and one may take $H=xy^2-x^3/3$. Thus characteristics are its level curves (a cubic foliation), and because $c=0$ the general solution is $u(x,y)=F(xy^2-x^3/3)$ for arbitrary <differentiable> $F$.