Problem: Find a uniformly valid approximation to the solution of $\epsilon y'' + (x^2-12)y' - y^3 = 0$ with boundary conditions $y(0)=1, y'(1)=1/2$ in the limit $\epsilon \ll 0+$. 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: $\boxed{y(x) \approx \left( 1 - \frac{1}{2\sqrt{3}} \ln\left( \frac{2\sqrt{3}-x}{x+2\sqrt{3}} \right) \right)^{-1/2} + \frac{\epsilon}{11} \left[ \frac{1}{2} + \frac{1}{11} \left( 1 - \frac{1}{2\sqrt{3}} \ln\left( \frac{2\sqrt{3}-1}{2\sqrt{3}+1} \right) \right)^{-3/2} \right] e^{-11(1-x)/\epsilon}}$