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