Say we have an infinite-order linear ODE of unknown $f$, of the following form: $$\sum_{n=0}^{\infty}a_n(x)f^{(n)}(x)=\gamma(x)$$ With the $(a_n)_{n\in\mathbb{N}}$ and $\gamma$ functions defined so that a solution exists.
Letting aside the technicalities to properly define this kind of equation for now, my question is, is there a way to deduce from the equation the asymptotic behavior of $f^{(n)}(0)$ as $n\rightarrow\infty$?
I know there are some countour integration based methods for asymptotics of $n$-th derivatives, but I'm not well versed into that enough to know if it's applicable to my case.
Any suggestion?