I want to find out if a generalisation of the methods found in this paper http://iopscience.iop.org/article/10.1088/1742-6596/788/1/012028/pdf have been developed to solve a system of ODEs only using parametric solutions?
For instance if there are a set of equations where $x_i(t)$ are some functions with $$ F(x_1,\ldots,x_n,t) =0 $$ and some $x_i'=G_i(x_1,\ldots,x_n,t)$ for $i = 1,\ldots,n$
Is there a technique (for instance seperation of variables like) to resolve such problems? Thanks!
Edit: I've seen similar two the usual two coupled system here Analytical solution to coupled nonlinear ODEs, but really if there is a known generalisation for three or more odes that is what I wish to discuss!
I'll give an example
Take the three body system $a(t),b(t),c(t)$, where $f(t),g(t)$ are known functions and we solve for $a,b,c$ with $$ a'(t) = f(t)c(t)$$ $$ b'(t) = g(t)c(t) $$ $$ c'(t) = \frac{1}{2}f(t) (1-2 a(t)) - g(t)b(t)$$
If one assumes the functions are non zero, then rearranging the first two we figure that the functions $a,b,c$ are constrained by
$$ c^2 + b(b-1) + a^2= \text{constant}$$
Which is an ellipsoidal constaint.