# Solving ODE with negative expansion power series

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?

• Cross-posted from math.stackexchange: math.stackexchange.com/questions/474959/… – Ricardo Andrade Aug 26 '13 at 4:10
• Hi dingo_d: A few comments. Zeroth, sorry you keep getting pinged around from site to site. First, some parts of the Stack Exchange network specifically encourage cross-posting, but between M.SE and MO it's frowned upon. As suggested on physics.stackexchange, it would be best to "migrate" it, which is something a moderator can do, I think. Second, as written the question is not well-suited here. There are at least two reasons. One is that it's not clear how your question is asking about "research level mathematics", so much as applying advanced mathematics to a specific physics calculation... – Theo Johnson-Freyd Aug 26 '13 at 5:31
• Two is more stylistic. You reference "the article" and "the authors", but don't supply a citation — you absolutely must do so here on MO, since this is part of basic academic writing (and MO does constitute a form of "academic" writing). Also, why do 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... – Theo Johnson-Freyd Aug 26 '13 at 5:36
• ...of what types of "solutions" you can hope for. A final third comment: I'm guessing, but not entirely sure, that $t$ is a "time" coordinate, $r$ and $\phi$ are "space" coordinates (so that you are imagining yourself to be in polar coordinates), and $g$ is a metric with components $g_{tt}$, etc.? Or is $g_{tt}=\partial_t^2 g$? Are there $g_{rt}$ components, etc.? I guess $\xi^\mu$ is a vector field? Not that it really matters — $g_{\mu\mu}$ might as well be $a_\mu$, and $(a_t,a_r,a_\phi)$ might as well be $(B,C,D)$, a trio of functions of three variables. But physical significance can ... – Theo Johnson-Freyd Aug 26 '13 at 5:41
• ...help to solve ODE. Oh, I guess I have a post-final fourth comment. Is the question to understand and work through the calculations formally applying the power series expansion around $r=\infty$ to calculate the formal asymptotics of $\xi(t,r,\phi)$ near $r=\infty$? Or to work through the analytics of the estimates necessary to confirm that those formal asymptotics are correct? Or...? (Sixth and really it: sorry for the extended comment.) – Theo Johnson-Freyd Aug 26 '13 at 5:43