Coercive estimates for a class of differential operators

A description is given of all matrix differential operators of the form $B\frac d{dt}+t^kA$, satisfying in $L_2(R^n)$ the coercive estimate
$$ \Bigl\|\frac d{dt}u\Bigr\|\le c\Bigl\|B\frac d{dt}u+t^kAu\Bigr\| $$
and the estimate
$$ \|u\|\le c\Bigl\|B\frac d{dt}u+t^kAu\Bigr\|. $$