Problem: Find the leading order WKB approximation for the specific differential equation: $$ \epsilon^2 y''(x) + (1+x^2) y(x) = 0 $$ with initial conditions at $y(0) = 1, y'(0) = 0$ 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_{WKB}(x) \approx (1+x^2)^{-1/4} \cos\left( \frac{1}{\epsilon} \left[ \frac{1}{2} x \sqrt{1+x^2} + \frac{1}{2} arcsinh(x) \right] \right)} $