Consider the following second order linear ODE,
$$L[g]=f(x)g''+b(x)g'+c(x)g=a,$$
where $a\neq 0$ is a constant and the coefficients can be as regular as you like. I am interested in conditions on the coefficient functions that ensure solutions are asymptotically stable. In the sense that $g \to 0$ as $x \to \pm \infty$. Such ODE's have been extensively studied over the past hundred years so I would be astonished if such a result wasn't available somewhere in the literature but yet I am having trouble finding such as result (of course you can find trivial conditions that ensure stability but I am looking for the most general available in the literature).
In the relatively recent article G. A. Grigorian, Stability criteria for second order linear ordinary differential equations. https://arxiv.org/abs/1905.06552 (2019), the author provides a characterisation of stability in the case where $a=0$, additionally one can use the bibliography given as a good survey of the work done over the past 100 hundred years.
The more general case when $a$ is in fact a function of $x$ has also been studied and sharp conditions provided but for a weaker type of stability, i.e the so called Hyers-Ulam Stability, for example see the paper https://www.researchgate.net/publication/220678382_Hyers-Ulam_Stability_of_Nonhomogeneous_Linear_Differential_Equations_of_Second_Order.
It seems to me a large proportion of the classical literature, by for example Richard Bellman in the 50's and 60's, is mainly focused on homogeneous equations.
With so much work done on such equations along with the vast literature on linear elliptic second order PDE's I find it hard to believe a good result isn't already out there, it seems I just don't know where to look! Any good references would be greatly appreciated!