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