Find the first three nonzero terms in each of two solutions (noy1t multiples of each other) xy''+y'-y=0

i know y1 and a_n(r) but i don't know how to differntiate a_n(r).

as i know, a_n(r)= 1/(r++n)

It looks like you're trying to solve the ODE via Frobenius' method. It's not immediately clear where the solution is to be centered, but I think we can safely assume it's to be taken about . Dividing through by , we get



and we observe that is a regular singular point.

Now, taking a solution of the form




and substituting into the ODE gives







The indicial polynomial admits only one root (of multiplicity two) at , so in fact a regular power series solution will exist. So in fact the ODE above reduces to



leaving us with the recurrence



We can solve this by recursively substituting the right hand side:



and so one of the solutions will be



Now the literature on the Frobenius method when the indicial polynomial has repeated roots suggests a second solution will take the form



but it's not clear to me *why* this works.