PDE - a few general questions

Hi, I just had a few questions on PDE's just to make sure I'm doing thigs right.

Question 1

Given a function f(x,y) is the integration with respect to one variable (say x) of zero c + g(y) where c is a constant?

Question 2

How woul you solve a question like u_y +u_xy = 0? I was thinking maybe you could devide by u_y but that seems way to easy...

Question 3

After finding the u(x,t) = F(x+ct) where c is a constant is the solution to a PDE, how would you apply the initial conditions? Am I looking to find a value for c?

Thank you!