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