# Asymptotic Solutions of Differential equation

Is there a way to get Mathematica to find the asymptotic solution, i.e. $r\rightarrow \infty$, of the following equation? It is unable to find the full solution. (a is a real number.)

 DSolve[-f''[r] - 1/r f'[r] + (Log[r]+a) f[r] == 0, f[r], r]


Just to be clear, the goal is to find analytic solutions, not numerical ones.

• Could you clarify what you mean by asymptotic solution? Do you mean $r\to\infty$? – Chris K Jul 11 '18 at 14:20
• Yes, that's right. – 121 Jul 11 '18 at 14:27

You can use AsymptoticDSolveValue to find the asymptotic approximation of f centered at a:
AsymptoticDSolveValue[-f''[r]-1/r f'[r]+(Log[r]+a) f[r]==0,f[r],{r,a,2}]

• @121 Yes, you can just replace a with Infinity, but AsymptoticDSolveValue is unable to return a result when a Log is included. If you replace Log[r] with r or 1/r then it will work. – Carl Woll Jul 11 '18 at 17:21
• You have hit the nail in the head as to why I was asking the question in the first place. With either $r$ or $1/r$, the full solutions are known. – 121 Jul 11 '18 at 19:32