I'm moving this here, as suggested from physics.stackexchange. The original is here.

So, I need to solve a system of ODE, using negative power expansion. I will give all the necessary equations and what's what here.

The ODE's are:

$\xi^r\partial_rg_{tt}+2g_{tt}\partial_r\xi^t=\mathcal{O}(r)$

$g_{tt}\partial_r\xi^t+g_{rr}\partial_t\xi^r=\mathcal{O}(1)$

$g_{tt}\partial_\phi\xi^t+g_{\phi\phi}\partial_t\xi^\phi=\mathcal{O}(1)$

$\xi^r\partial_rg_{rr}+2g_{rr}\partial_r\xi^r=\mathcal{O}\left(\frac{1}{r^3}\right)$

$g_{rr}\partial_\phi\xi^r+g_{\phi\phi}\partial_r\xi^\phi=\mathcal{O}(1)$

$\xi^r\partial_rg_{\phi\phi}+2g_{\phi\phi}\partial_\phi\xi^\phi=\mathcal{O}(r)$

where

$g_{tt}=-\left(1+\frac{r^2}{l^2}\right)$, $l^2$ is a parameter, not depending on the variables (a constant)

$g_{rr}=\left(1+\frac{r^2}{l^2}\right)^{-1}$, and

$g_{\phi\phi}=r^2$.

I need a solution for $\xi^\mu$, where $\mu=t,r,\phi$. I should assume the solution in the form of

$\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$, where this should be viewed as an expansion in $1/r$ (expansion around $r=\infty$), and we need to assume that the each series truncates for some large N onward different for each component.

I put this in the first equation and get this

$\frac{2}{l^2}\sum\limits_n\xi^r_nr^{n+1}+2\sum\limits_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum\limits_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$

the comma means derivative ($\partial_t\xi^t_n\equiv \xi^t_{n,t}$).

How do I expand this? In the article the author gets:

$\xi^r_{n-1}+l^2\xi^t_{n,t}+\xi^t_{n-2,t}=0,\ n\ge 2$

$l^2(n+1)\xi^r_{n+1}+(n-2)\xi^r_{n-1}=0,\ n\ge 2$

$\xi^r_{n-1}+\xi^\phi_{n-2,\phi}=0,\ n\ge 2$

$l^4(n+1)\xi^t_{n+1}-l^4\xi^r_{n,t}+2l^2(n-1)\xi^t_{n-1}+(n-3)\xi^t_{n-3}=0,\ n\ge 3$

$l^2\xi^\phi_{n-2,t}-l^2\xi^t_{n,\phi}-\xi^t_{n-2,\phi}=0,\ n\ge 1$

$l^2(n-1)\xi^\phi_{n-1}+(n-3)\xi^\phi_{n-3}+l^2\xi^r_{n,\phi}=0,\ n\ge 3$

I cannot get to this part :\

Do I need to expand to negative n, and then cancel out the $\mathcal{O}(r^n)$ terms? How to do this?

whydo you "need to solve a system of ODE, using negative power expansion"? Homework (if so, not appropriate here)? Understanding an article? Something else? Being able to answer this well help to communicate what kind of "solve" you're going for — for a generic system of ODE, there's not much you can say, but for specific systems, there's a range... $\endgroup$ – Theo Johnson-Freyd Aug 26 '13 at 5:36can... $\endgroup$ – Theo Johnson-Freyd Aug 26 '13 at 5:41