Let $S(\eta)$ be a function that verifies the equation
$$S'''-\eta S' = H (\eta),$$
where $H(\eta)=a Ai (\eta) + b Gi(\eta) + c Ai'(\eta) + d Gi'(\eta) + e Ai''(\eta) + f Gi'' (\eta)+ g Ai'''(\eta) + h Gi''' (\eta)$. The function $Gi(\eta)$ is defined as
$$Gi(\eta)=Ai(\eta) \int_{\eta_0}^\eta Bi(\eta)d\eta - Bi (\eta) \int_\infty^\eta Ai(\eta) d\eta$$
and $a,b,c,d,e,f,g,h$ are given constants. The variable $\eta$ is complex. How could we solve for $S(\eta)$? All I know is that $S$ can be put as a function of $Ai(\eta)$, $Gi(\eta)$ and its derivatives.