Problem: Find a uniformly valid approximation to the solution of $\epsilon y'' - (x+1) y' + x^2 + x + 1 = 0$ with boundary conditions $y(0) = 1, y(1) = 2$ in the limit $\epsilon \ll 0+$. 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: $\boxed{y = \frac{1}{2} x^2 + \ln{(x+1)} + 1 + (\frac{1}{2} - \ln{2}) e^{-2(1-x) / \epsilon}}$