For functions $a(x)$ and $b(x)$ and "sources" $S_1(f,g)$, $S_2(f,g)$ and $S_3(f,g)$ lets say one has the differential equations for functions $f(x)$ and $g(x)$,

$f' + af + bg = S_1(f,g) + S_2(f,g)$

$g' + f = S_3(f,g)$

Now if $S_1= S_2 = S_3 = 0$ i.e all the sources are turned off then the system reduces to the differential equations, $g'' + ag' - bg = 0$ (by eliminating $f$ as $f = - g'$). For this 2nd order ODE for g let $G_g$ be the Green's function.

And let $f_{13}(x)$ and $g_{13}(x)$ be the solutions to the system when $S_1$ and $S_3$ are on but $S_2 = 0$.

(..if $f_0(x)$ and $g_0(x)$ are the solutions with all sources turned off then $f_{13}$ and $g_{13}$ can be thought to be given as a power-series in $f_0$ and $g_0$ respectively..)

Similarly when sources are all turned off, one can eliminate $g$ between the two equations to get a 2nd order ODE for f as, $f'' + (a b'/b)f'+(a'-(ab')/b-b)f=0$ and then let $G_f$ be the Green's function for this.

Given so much of data $G_g$, $G_f$, $f_{13}$ and $g_{13}$ can one build the full solutuions when all the 3 sources are on?

Will it be an approximation (how good?) if one says that the full $g$ is given by,

$g (x) = g_{13}(x) + \int _{0}^x G_g(x,x')S_2 (f_{13},g_{13}) dx'$

and similarly can one write (approximate?) that,

$f(x) = f_{13}(x) + \int _0 ^x G_f(x,x')S_2(f_{13},g_{13})dx'$

(..I guess that in the above the "=" should be replaced by "~" and one can pertrubatively develop a solution by taking the $g$ or $f$ obtained in the first pass and then using that again as the new $f_{13}$ and $g_{13}$ respectively..though such an iteration is a bit hard to make sense of given that $f_{13}$ and $g_{13}$ themselves are given as an infinite power-series - so it seems like for any truncation of that power-series one would have to still do infinite updates here!..)