I would like to change the coordinate system from $t$ to $x$ in a way that operator $t+\frac{d}{dt}$ becomes $\frac{d}{dx}$ or $\frac{d^2}{dx^2}$. Is there any general procedure for this type of problem?
I have managed to solve a similar problem as follows. Consider operator $t^2\frac{d}{dt}$ for $t>0$. By setting $\frac{d}{dx}=t^{2}\frac{d}{dt}$, we have $\frac{dt}{dx}=t^2$, therefore $\frac{dx}{dt}=\frac{1}{t^2}$ and $x(t)=\frac{-1}{t}+constant$.
You can't do it with a change of variables, however note that $$ e^{-t^2/2}\frac{d}{d t} e^{t^2/2}=t+\frac{d}{dt} $$