If $f$ is a polynomial, how does $f(\frac{d}{dt})$ act on $y$? If $f\left(\frac{d}{dt}\right)=a_n\frac{d^n}{dt^n}+\dots+a_1\frac{d}{dt}+a_0$, then whether
$$f\left(\frac{d}{dt}\right)(y)=a_n\frac{d^n}{dt^n}y+\dots+a_1\frac{d}{dt}y+a_0y$$
or
$$f\left(\frac{d}{dt}\right)(y)=a_n\frac{d^n}{dt^n}y+\dots+a_1\frac{d}{dt}y+a_0$$
I am confused.

The correct answer  is $$(\frac{d}{dt})(y)=a_n\frac{d^n}{dt^n}y+\dots+a_1\frac{d}{dt}y+a_0y$$
Otherwise you lose the linearity of your operator.
