I am not quite sure if this question is appropriate for this site as it might be not of a research level. 
I am interested in the following ordinary differential equation on the real line
$$f’’(x)+(x-a)f(x)=0$$
where $a>0$. 



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> Is it possible to solve it explicitly? or using say hypergeometric functions? In any case, I would be interested to know asymptotic behavior of solutions when $x\to +\infty$. 
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Asymptotic behavior of a solution of an ODE
It is possible to solve it explicitly in terms of the Airy function.
Airy's equation in the standard form is
$$y''=xy.$$
Your equation is reduced to this by $x\mapsto-x$ followed by a shift of the independent variable.
Airy functions have been thoroughly studied and almost everything you want to know about them is known. In particular there is a full asymptotic expansion for them as $x\to\infty$. Your question is about the behavior of Airy function as $x\to-\infty$. It oscillates, has infinitely may zeros, and the shape of this oscillation is very well described in the special function handbooks.
Airy function can be obtained from a hypergeometric one by a confluence process.
 Type "Airy functions" on Google.


