I'm having trouble finding the recurrence relation of the following non linear differential equation:
$y''(x)+p(x)y'(x)+y^2(x)=0$
with $y(0)=1$
and $y'(0)=0$
where $p(x)=\sum_{l=0}^{inf}p_{l}x^{l}$
Using the usual power series solution form $y=\sum_{0}^{inf}a_{n}x^{n}$
$y'=\sum_{0}^{inf}a_{n}nx^{n-1}$ and $y''=\sum_{0}^{inf}a_{n}(n)(n-1)x^{n-2}$ I get:
$\sum_{0}^{inf}a_{n}(n)(n-1)x^{n-2}+\sum_{0}^{inf}a_{n}nx^{n-1} * \sum_{l=0}^{inf}p_{l}x^{l} + \sum_{0}^{inf}a_{n}x^{n} * \sum_{0}^{inf}a_{n}x^{n}$
How do I simplify the Cauchy product? Would this work?
$\sum_{0}^{inf}a_{n}(n)(n-1)x^{n-2}+\sum_{n=0}^{inf}(\sum_{0}^{n}a_kp_{n-k})x^{n-1}+\sum_{n=0}^{inf}(\sum_{0}^{n}a_ka_{n-k})x^{n}$
Since equalizing the powers/indices is crucial if I want to find the recurrence relation, I just want to make sure I got the right ones. I'm especially not sure about the middle term with $x^{n-1}$ and $p^{l}$, all the examples I found on the net were products of $\sum a_nx^n$ and $\sum b_nx^n$
Also, I'm not even sure about the method. If there is a simpler way to find the recurrence relation, it would be perfection to be offered awareness of its existence.
Any hint would be immensely appreciated. Sincerely, Tinky Winky