Closed form for a differential recurrence

Question

Function $d_0$ is given, $d_{n+1}(y)=d_n'(y)d(y)$ for every $n\in\mathbb{Z}$ (apostrophe means differentiation).

Can we write a closed formula (or at least an explicit sum) for:

1. $d_n(y)$?

2. $d_0(y)+d_1(y)t+d_2(y)t^2+\dots+d_n(y)t^n$?