$$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x)\Rightarrow$$
$$y'(x)=\frac{{a_1}}{x} \left(\left({\lambda}+x^2\right) M(-\frac{{\lambda}}{2},\frac{1}{2},x^2)-({\lambda}-2) M(1-\frac{{\lambda}}{2},\frac{1}{2},x^2)\right)+2 {a_2} {\lambda} H({\lambda}-1,x)$$

for $\lambda$ negative non-integer and $x\rightarrow-\infty$ the $a_1$ term diverges as $e^{x^2/2}$, while the $a_2$ diverges as $e^{x^2}$, so it does not seem possible to cancel the divergence by a suitable choice of $a_1$, $a_2$.