Problem: Find the leading order WKB approximation for the specific differential equation: $$ \epsilon^2 y''(x) - (1+\cos{x}) y(x) = 0 $$ with initial conditions at $y(0) = 0, y'(0) = \frac{2^{5/4} \cosh{(1)}}{\epsilon}$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1$). 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: $\boxed{y(x) \sim 2\cosh(1) (1+\cos x)^{-1/4}\sinh\left[\frac{2\sqrt{2}\sin(x/2)}{\epsilon}\right]}$