Problem: Find a uniformly valid approximation to the solution of $\epsilon y''(x) - y'(x) + e^{y(x)} = 0$ with boundary conditions $y(0) = -3, y(1) = 0$ 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(x) = -\ln(e^{3}-x) + \ln(e^{3}-1)e^{\frac{x-1}{\epsilon}}} $