Problem: Find a uniformly valid approximation to the solution of $ \varepsilon y'' + \left(1 + \frac{2\varepsilon}{x} - \frac{2\varepsilon^3}{x^2}\right) y' + \frac{2y}{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: $\boxed{y(x) = 1 + \left( x^{-2} + 2\varepsilon(x^{-3} - x^{-2}) - 1 \right) e^{-2\varepsilon^2 / x}}$