$\displaystyle u_x-3u_y=0$

$\displaystyle u(x,x)=x^2$

$\displaystyle \omega_{\xi}(\cos{\alpha}-3\sin{\alpha})-\omega_{\eta}(\sin{\alpha}+3\cos{\alpha})=0$

$\displaystyle \displaystyle\cos{\alpha}=\frac{1}{\sqrt{10}} \ \ \ \sin{\alpha}=\frac{-3}{\sqrt{10}}$

$\displaystyle \displaystyle\sqrt{10}\omega_{\xi}=0\Rightarrow\om ega_{\xi}=0$

$\displaystyle x=\xi+B\eta \ \ \ y=-3\xi+D\eta$

$\displaystyle \displaystyle\int\omega_{\xi}d\xi=\int 0d\xi\Rightarrow\omega(\xi,\eta)=g(\eta)$

I am lost with initial condition and what to set D and B equal too.