The problem is (book : ordinary differential equations, Birkhoff, 1969, exercises C.6, p.72):
a) Obtain a recursion relation on the coefficient $a_k$ of power series solutions $\sum a_k x^k$ of Pearson's DE $y' = (D+Ex)y/(A+Bx+Cx^2), A ≠ 0$.
b) What is the radius of convergence of the solution ?
c) Integrate by quadratures, and compare.
For a), i found $a_{k+1} = \frac{a_{k-1}(E-C(k-1))+a_k(D-Bk)}{A(k+1)}, k \geq 2$
For b), first, i wanted to use $\lim \left| \frac{a_{k+1}}{a_k} \right|$, but i think it's not a good idea. I tried to define a new sequence $(b_n)$ : $b_k = a_{k+1} - a_k$, but i'm lost...
For c), i have $\int \frac{D+Ex}{A+Bx+Cx^2}dx = \frac{1}{2C} \left( E \ln(A+x(B+Cx)) - \frac{2(BE-2CD)}{\sqrt{4AC-B^2}}\arctan(\frac{B+2Cx}{\sqrt{4AC-B^2}}) \right)$, but what can i say ?...
Hint.
Regarding the convergence radius we can consider sufficient conditions. Observing the recurrence
$$ a_{k+1}=\frac{E-C(k-1)}{A(k+1)}a_{k-1}+\frac{D-Bk}{A(k+1)}a_k $$
for big $k$ we have the recurrence
$$ Aa_{k+1}+B a_k+C a_{k-1}=0 $$
This recurrence has as solution
$$ a_k = \left(\frac{-B+\sqrt{B^2-4AC}}{2A}\right)^k c_1+ \left(\frac{-B-\sqrt{B^2-4AC}}{2A}\right)^k c_2 $$