Help me please. How to find the maximum area of the uniqueness of the Cauchy problem $$u_{xx}-u_{xy}-2u_{yy}+3u_x-6u_y=0,$$ $$u|_{x=0, y>0}=e^{-y},\\ u_x|_{x=0, y>0}=e^{-y}.$$
I found the solution by changing variables from the characteristic equation $dy^2+dxdy-2dx^2=0$ and canonicalization. $$u_{\psi\phi}+u_{\phi}=0,\; \phi=y-x,\; \psi=y+2x.$$ $$u(x,y)=e^{-y-2x}(3x+1).$$ But I can't figure out how to find this area from the characteristics.
It's true that this is the area $\{(x,y):y-x>0,y+2x>0\}$? That is an area where, through any point are two characteristic lines intersect $\{(0,y):y>0\}$.