Explicit solutions to this nonlinear system of two differential equations I am interested in a system of differential equations that is non-linear, but it doesn't seem to be too crazy. I'm not very good at non-linear stuff, so I thought I'd throw it out there.
The actual equations I'm looking at have several parameters that'd I'd like to tweak eventually.
q' = k - m / r
r' = i - n r - j q

i, j, k, m and n are all real-valued constants. I'm guessing that this system would be cyclical in nature, but I'm not sure if it has any explicit solution, so I have produced a version of it with the constants removed to see if that can be solved:
q' = 1 - 1 / r
r' = 1 - r - q

Anyone know if either of these are solvable and what kind of techniques would be needed to solve them if so?
The first equation is based on a polar coordinate system where Q (or theta) is the angle and r is radius, and I've made a number of simplifications to make it somewhat tractable.

Taking that second question,
$r' = i - nr - jq$
and differentiating gives
$r'' = -nr' - jq' = -nr' - j(k-\frac{m}{r})$
or in other words
$r'' + ar' + \frac{b}{r} = c$
which is a much simpler differential equation only one variable. I think that you could probably solve this with power series or clever guessing, but it needs to be worked out.
