Equations of motion for the Schwarzschild metric (nonlinear PDE)

[BiGyElLoWhAt]

TL;DR Summary

Solution to equation of the form d^2r/dl^2 + f(r)(dt/dl)^2 + g(r)(dphi/dl)^2 where l is an arbitrary parameter. Working in spherical coordinates

I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance, most of the things I know to try from ODE don't look like they'll work.

$\frac{d^2r}{d\lambda^2} + 1/2 \frac{r_s(r-r_s)}{r^3}(\frac{dct}{d\lambda})^2 +1/2 \frac{r_s}{r(r_s-r)}(\frac{dr}{d\lambda})^2 +(r_s-r)(\frac{d\phi}{d\lambda})^2 =0$
Any pointers, solution methods, etc would be greatly appreciated.


**crosses fingers for good latex render because preview is broken**


Edit**
Having stepped away and played a video game for 45 or so minutes, I'm fairly certain that my other solutions are relevant here, as I effectively have solutions for $\frac{dct}{d\lambda}$ and $\frac{d\phi}{d\lambda}$

I have:
$\frac{dct}{d\lambda} = \frac{Ar}{r-r_s}$
and
$\frac{d\phi}{d\lambda} = \frac{B}{r^2}$
I'm pretty sure the constant A has to to with total energy and B has to do with angular momentum in the phi direction, beyond that I'm not really sure what those 2 are at the moment.
Plugging that into what I have for my dr equations:

$\frac{d^2r}{d\lambda^2} + 1/2 \frac{r_s(r-r_s)}{r^3}[\frac{Ar}{r-r_s}]^2 +1/2 \frac{r_s}{r(r_s-r)}(\frac{dr}{d\lambda})^2 +(r_s-r)[\frac{B}{r^2}]^2 =0$
I haven't gotten it down on paper, just did some copy paste. I'm going to continue to stare at this paper and see if anything pops out at me, but still (obviously) feel free to jump in at any time and give me some tips.

Edit** ctd.
So I cleaned it up a bit, but am not really sure what to do now. This is what I have:
$\frac{d^2r}{d\lambda^2} + 1/2 \frac{r_s}{r(r_s-r)} (\frac{dr}{d\lambda})^2 = -A^2/2 \frac{r_s}{r(r-r_s)} - B^2\frac{r_s - r}{r^4}$
So for the other 2 equations, I had homogeneous equations, and was basically able to do something like $\frac{d}{d\lambda}[f(r)\frac{dct}{d\lambda}]$ or with dphi for the phi equations. This almost looks like I should be able to do that, but I can't easily rearrange to get a condition on f(r) (before i had f'/f = the function attached to the first order term), and 2nd, I'm not sure how to integrate a random function of r w.r.t. lambda (rhs) where r is a function of lambda.

--this really reminds me of the drag equation (nonlinear drag)

 

[gtoguy97]

where you have a drag term on the rhs, and I think it's related to the "coupling" terms in the equation of motion.Anyways, I'm still stuck and figured I'd update this post with more information. Any help is appreciated!