Problem: Find the leading order WKB approximation of $\epsilon^2 y'' = (1 + x \sin(x)) y$ subject to boundary conditions $y(0) = 0$, $y(1)= 1$ in the limit $\epsilon \rightarrow 0^+$. Approximate any integrals with a polynomial in $x$, up to third order in $x$. Put your final answer in a LaTeX \boxed{} environment.
Solution: $\boxed{y = (1 + \sin(1))^{1/4} \left(\sinh\left(\frac{7}{6\epsilon}\right)\right)^{-1}(1 + x\sin(x))^{-1/4} \sinh\left(\frac{(x + x^3/6)}{\epsilon}\right)}$