Problem: Find a uniformly valid approximation to the solution of $\epsilon y''(x) + y'(x) = -e^{-x}$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
Solution: $\boxed{y(x) = e^{-x} + 2 - e^{-1} - (2 - e^{-1})e^{-x/\epsilon}}$