After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following:
Given the PDE
$\dfrac{\partial f}{\partial x} = 0 \quad $ on $U = { (x,y) \in \mathbb R^2 ; y>0, 1 < x^2 + y^2 < 4}$
I am looking for a solution $f$ that does not only depend on $y$.
How can this be?!
The exercise is taken form Lee's "Introduction to smooth manifolds", p. 517 at the end of the chapter on the Frobenius theorem.
(Note: According to the errata, the condition on $U$ is $y > 0$, not $x > 0$ as your copy of the book might state).
Thanks in advance!
S. L.