This question occured to me while trying to solve a three-body-esque problem involving three charged particles placed along a straight line (ignoring effects of electromagnetic radiation etc. etc.). Anyway, for particles of masses $m$, $2m$ and $5m$ and charges $q$, $q$ and $2q$, respectively, we obtain this system of differential equations:
$\begin {cases}2{m\over{kq^2}} {d^2{r_1}\over{dt^2}} = {3\over{r_1^2}} + {4\over(r_1 + r_2)^2} - {2\over{r_2^2}}\\ 10{m\over{kq^2}} {d^2{r_2}\over{dt^2}} = -{5\over{r_1^2}} + {4\over(r_1 + r_2)^2} + {14\over{r_2^2}}\end {cases}$
(where $r_1$ and $r_2$ are the distances between $m$ and $2m$ and between $2m$ and $5m$, respectively, as functions of time)
It is known that a particular solution can be obtained for $r_1$ = $r_2$. What I am interested in, however, is whether a general solution can be obtained now that we have a particular solution, like in the method of variation of parameters for a second order linear ODE?