Problem: Find the leading order behavior of $\epsilon y'' = e^xy$ subject to boundary conditions $[y(0) = 0, y'(0) = 1]$ in the limit of $x\to\0^+$.
Solution: $\boxed{y(x) = \sqrt{\epsilon} e^{-\frac{x}{4}}\sinh(\frac{2e^{\frac{x}{2}}-2}{\sqrt{\epsilon}})}$