For the PDE $u_y + u u_x = 1$, sketch a plot of $\Gamma$ and a few representative curves, including the envelope curve. Conditions: $u=0$ on the curve $y^2=2x$, and $y,x>0$. Express $u$ as a function of $x,y$.
I'm new to PDE's and very lost. I saw similar problems on here, but not identical. I'm pretty sure I have to use the Method of Characteristics. But I'm not sure how you find the curve Gamma, or parameterize everything.