Problem: Find the leading order approximation of $y(x)$ from the differential equation $\epsilon^2 y''(x)=(\sin x) y$ subject to boundary conditions $y(\frac{\pi}{2}) = 1, y'(\frac{\pi}{2}) = 0$ in the limit of $\epsilon \to 0$. The answer should be in terms of the incomplete beta function $B_z(a, b)$ where $B_z(a, b)=\int_0^z t^{a-1}(1-t)^{b-1} d t, 0 \leq z \leq 1$
Solution: $\boxed{\frac{1}{(\sin x)^{\frac{1}{4}}} \cosh \left(\frac{1}{2 \epsilon}\left[B_{(\sin(x))^2}\left(\frac{3}{4}, \frac{1}{2}\right)-B\left(\frac{3}{4}, \frac{1}{2}\right)\right]\right)}$.$