Some very nice tricks there to get rid of the explicit variable. I get the same result, although I grouped differently because I saw the same pattern repeating:
$$ ( \ddot{g}+ \dot{g} ) + 2(\dot{g} +g) + \frac{(\dot{g} +g)^2}{1-g} + \frac{g(\dot{g} +g)}{1-g}=0 $$
Looks so tantalizingly...
Hmmm... interesting. Not sure if it's what you intended, but gets me thinking there might be a coordinate transform that would result in a DEQ that's easier to solve even if it looks more complicated. (Yes, this is from a GR metric I've been playing with in my spare time :) ) Though I'm not...
A geometry problem I'm working on has boiled down to finding a function ##f(t)## such that $$f'' + \frac{2}{t}f' + \frac{f'^2}{\left( 1 - \frac{f}{t} \right) t } + \frac{f'f}{\left(1- \frac{f}{t} \right) t^2} = 0$$ It has two fairly simple solutions, namely ##f(t) = a## and ##f(t) =...