Changing coordinates back for a differential equation

Question

I change the variable $(x, y) \to (\xi, \mu), \xi = x+y, \mu = x$ to transform $u_x-u_y=u$ to $u_{\mu} = u$ so that $u(\xi, \mu) = Ce^{\mu}. $ How do I change coordinates back? It may be something simple but I can't see a way. I'm supposed to get $u(x,y) = Ce^x f(x+y)$.

Answer 1

You forgot that when you integrate in $\mu$, your integration constant depends on $\xi$. This yields $$ u=e^{\mu}C(\xi) $$ Now plugging back in, $\mu=x$ and $\xi=x+y$ yields the general solution $$ u(x,y)=e^xC(x+y) $$