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