# Asymptotic behavior of a solution of an ODE

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 $a>0$.

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 $x\to +\infty$.

• Maple performs its general solution $f \left( x \right) ={\it \_C1}\,{{\rm Ai}\left(-x+a\right)}+{\it \_C2} \,{{\rm Bi}\left(-x+a\right)}$ in terms of the Airy functions (see maplesoft.com/support/help/Maple/… for more info). Maple also finds its asymptotics at $\infty$. I think allthat is well known. Indeed, this is not a question at the research level. – user64494 May 14 '18 at 15:03
• Up a time translation and inversion, $t=-x+a$ it is Airy's equation $u''(t)+tu(t)=0$, whose 2 linearly independent solutions (with initial data $(u(0),u'(0) = (0,1)$ resp $(1,0)$ ), have a very simple entire power series expansion. Check en.wikipedia.org/wiki/Airy_function for info, included the asymptotics. – Pietro Majer May 14 '18 at 15:32
• For the record: WolframAlpha also does the job, free of charge. – Mateusz Kwaśnicki May 14 '18 at 17:34
• @Mateusz Kwaśnicki: But not the required asymptotics. Free WA is not very strong. – user64494 May 14 '18 at 17:39
• @user64494: Of course, you are right. On the other hand, it is not as bad as it seems. And once you know the name of the solution, it is quite straightforward to look up the property you need. – Mateusz Kwaśnicki May 14 '18 at 17:48

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.