Consider second-order ordinary differential equations of the form

$u''(t)=a(t)u(t)-2$

I'm interested in general criteria on the function $a(t)$, which guarantee respectively rule out the existence of a positive solution $u(t)>0$, when $a(t)$ is defined on an interval $]b,\infty[$. I believe that under suitable regularity conditions the long-term behaviour of $a(t)$ is crucial and some kind of transition occurs.

As an example, for $a(t)=c/t^2$ with a fixed constant $c \in ]0,\infty[$, WolframAlpha gives a general complex solution for $c \neq 2$ and I guess that a positive solution $u(t)$ exists for $c>2$ but not for $c<2$. Even if this is not correct, this is the kind of phenomena I'm interested in for $a(t)$ as general as possible. Any ideas?

I've taken a look at some research papers on positive solutions of differential equations and related problems, but unfortunately I didn't find anything which fits exactly. Any literature suggestions?