Let $S(\eta)$ be a function that verifies the equation
$$S'''-\eta S' = H (\eta),$$
where $H(\eta)$ is given by \begin{equation}\label{h_st} H=\sum_{n=0}^{n=3} \left [ a_n Ai^{(n)} + g_n Gi^{(n)} \right ], \end{equation}
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 $a_n,g_n$ are given constants. The variable $\eta$ is complex. It is useful to note the set of relations \begin{equation}\label{recurrence_st} \begin{aligned} &\mathcal{L}Ai=0, \quad \quad \mathcal{L}{Ai}'=Ai, \quad \mathcal{L}{Ai}''=2 {Ai}', \quad \cdots, \quad \mathcal{L}Ai^{(n)}=n Ai^{(n-1)};\\ &\mathcal{L}Gi=1/\pi, \quad \mathcal{L}{Gi}'=Gi, \quad \mathcal{L}{Gi}''=2 {Gi}', \quad \cdots, \quad \mathcal{L}Gi^{(n)}=n Gi^{(n-1)}, \end{aligned} \end{equation} where $\mathcal{L}=d^2/d\eta^2-\eta$ is the Airy operator.
How could we solve for $S(\eta)$? I have tried variation of parameters but I obtained a very difficult expression and I have been told that the solution has the same form as $H$,
\begin{equation} S=\sum_{n=0}^{n=3} \left [ \tilde{a}_n Ai^{(n)} + \tilde{g}_n Gi^{(n)} \right ], \end{equation}
what's the relation between $\tilde{a}_n, \tilde{g}_n$ and $a_n,g_n$?