Problem: Find a uniformly valid leading order approximation to the solution of $$\epsilon y'' - y' = \sin(\pi x)$$ with boundary conditions $ y(0) = 0, y(1) = 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: $$\boxed{y(x) = \frac{\cos(\pi x) - 1}{\pi} + \frac{2}{\pi}e^{\frac{x-1}{\epsilon}}}$$