# 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 $r\to\infty$? 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}]


(-a + r - (-a + r)^2/(2 a)) C[2] + C[1] (1 - 1/2 (-a + r)^2 (-a - Log[a]))

• Could you explain what you mean by an "asymptotic approximation"? Instead of a, could you have found the solution around infinity, analogous to how Taylor expansions can be performed around infinity?
– 121
Jul 11 '18 at 14:56
• @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. 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