The recurrence relation is $$ y_n = d_1 y_{n-1}+\frac{d}{dx}[y_{n-1}] $$ a good thing to note as well is $$ \frac{d}{dx}[d_n] = d_{n+1} $$
This is terrible to expand out after a good while, The main goal is to represent this as a summation or as a function of some sort. I would rather officially solve myself So tips would be ideal, but answers are great too.