Problem: Find a uniformly valid approximation to the solution of $\epsilon y'' + (\ln x)y' - x(\ln x)y = 0$ with boundary conditions $y(1/2) = 1, y(3/2) = 1$ in the limit $\epsilon \to 0$. 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: $\boxed{\frac{1}{2} \left( e^{-\frac{1}{8} + \frac{x^2}{2}} + e^{-\frac{9}{8} + \frac{x^2}{2}} \right) + \frac{1}{2} \left( e^{-\frac{9}{8} + \frac{x^2}{2}} - e^{-\frac{1}{8} + \frac{x^2}{2}}\right) * erf\left(\frac{x-1}{\sqrt{2\epsilon}}\right)}$