Problem: Find a uniformly valid leading order approximation to the solution of $\epsilon y'' + (e^x - 1) y' = 0$ with boundary conditions $y(-1) = 0$, $y(1) = 1$ 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: $\boxed{y_{unif}(x, \epsilon) = \frac{1}{2} \text{erf}\left(\frac{x}{\sqrt{2\epsilon}}\right) + \frac{1}{2}}$