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$$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x)\Rightarrow$$ $$y'(x)=-2a_1\lambda x M(1-\tfrac{1}{2}{\lambda},\tfrac{3}{2},x^2)+2 {a_2} {\lambda} H({\lambda}-1,x)$$ The asymptotics for $x\rightarrow-\infty$ and $\lambda$ negative non-integer is $$y'(x)\rightarrow\frac{2 e^{x^2}(-x)^{-\lambda} }{\sqrt{\pi }\, \Gamma \left(-\lambda/2\right)}\bigl(a_2 \Gamma \left(-\lambda/2\right) \Gamma (\lambda+1)\sin \pi \lambda -\pi a_1\bigr).$$ So this vanishes if $$a_2 \Gamma \left(-\lambda/2\right) \Gamma (\lambda+1)\sin \pi \lambda =\pi a_1.$$