$\def \pux{ {\partial u \over \partial x}} \def \puy{ {\partial u \over \partial y}} \def \px{ {\partial \over \partial x}} \def \pz{ {\partial \over \partial z}} \def \py{{ \partial \over \partial y}}$ In Lee's Introduction to Smooth Manifolds text, problem 9-22.c is to find a solution to the quasilinear Cauchy problem:
$$ \pux + u\puy = y, \qquad \text{ with } u(0, y) = 0$$
Using the method of characteristics, I found the solution:
$$u(x,y) = y\left( {e^x - e^{-x} \over e^x + e^{-x}} \right)$$
Problem 9.23.c is to show that 9.22.c has no global solutions. The solution for $u$ above seems valid $\forall x,y\in \Bbb R^2$.
What am I missing?
You're right -- this problem was incorrectly stated in the book. There's a corrected version in my correction list. (It would probably be a good idea to download that list and keep it nearby while you read!)
Sorry for the confusion.