Problem: Find a uniformly valid leading order approximation to the solution of $$ \epsilon y'' + 2y' + y = \cos\left(\frac{\pi x}{2}\right)$$ with boundary conditions 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: $$ \boxed{y = \frac{1}{1+\pi^2}\left(\cos\left(\frac{\pi x}{2}\right)+\pi\sin\left(\frac{\pi x}{2}\right)\right) - \frac{\pi \sqrt{e}}{1+\pi^2} e^{-x/2} + \frac{\pi(1+e)}{1+\pi^2} e^{-2(x+1)/\epsilon}} $$