I know that the general form solution to the Hermite differential equation $$ y''-2xy'+2\lambda y=0$$ is $$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x),$$ where $M(\cdot,\cdot,\cdot)$ is a confluent hypergeometric function of the first kind, and $H(\cdot,\cdot)$ is a Hermite polynomial.

For a general value of $\lambda$ (negative and non-integer real valued), is there a special solution to the Hermite differential equation such that it's first order derivative goes to zero for $x\rightarrow-\infty$? In other words, is there a parametric characterization of $a_2$ as a function of $a_1$ and $\lambda$ such that $y'(x)\rightarrow 0$ as $x\rightarrow-\infty$?

From mathematica numerical calculations it seems that such a special case exists. However, I was not able to characterize it explicitly. I would appreciate any help on this. Thank you!