Problem: Find a uniformly valid approximation to the solution of $\epsilon y''-(1+2x^2)y+2=0$ with boundary conditions $y(0)=y(1)=1$ 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: $\boxed{y(x)=\frac{2}{1+2x^2}-e^{-\frac{x}{\sqrt{\epsilon}}}+\frac{1}{3}e^{\frac{\sqrt{3}(x-1)}{\sqrt{\epsilon}}}}$