I would like to approximate the solution of
$$\epsilon y''(x)+ x^2 y'(x) - \lambda y(x) = 0 $$
with boundary conditions $y'(0)=0$ and $y(\infty)=0$. The parameter $\epsilon$ is small.
To approach this problem, I have attempted to use a boundary layer approach. I consider that near $x=0$, the middle term is subdominant, so that $y_I(x) = A \cos ( \sqrt{\lambda}x).$ Likewise, for $x\rightarrow \infty$, I consider that the highest order term can be neglected, so that $y_O(x) = B \exp(-2\lambda/x)$.
Now each of these solutions matches its associated boundary condition -- $y'(0)=0$ for $y_I$, and $y(\infty)=0$ for $y_O$. However, the solution still contains two unrestrained constants.
How can I use the inner and outer solutions to solve for $A$ and $B$ and obtain an approximation for $y(x)$?