Crash course in pde's

I'm working in a problem involving some partial differential equations (of nonlinear type), and as I know very little about pde's, I've decided to commit myself to a "crash-course", trying to read and do the exercises of Lectures on Partial Differential Equations, by V. I. Arnold. I did something similar (though it was a course I had, a full semester) with Arnold's Mathematical Methods in Classical Mechanics, and I got a grasp of mechanics this way (but I've forgotten a great deal, and I skipped some exercises). I will try to finnish it within next week.

PS: I'm stuck at problem 2, after spending almost an hour with problem 1. And this is page 3 actually. Oh my...
Written by Ruben Berenguel