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Given the following ode for $x\rightarrow\infty$:

$$\left(x f(x)^3 \left(\frac{(x f^\prime)^\prime}{x}\right)^\prime\right)^\prime=0,$$

in the sense of "asymptotics", the equal sign is equivalent to "$\sim$".

Assume the function $f(x)>0$ defined on $x>0$. Is it possible to obtain a symbolic solution with AsymptoticDSolveValue?

AsymptoticDSolveValue[D[x*f[x]^3*D[D[x*f'[x], x]/x, x], x] == 0, f[x], {x, \[Infinity], 2}]

However, the above naive code produces an output the same as the input.

My questions:

  1. Can AsymptoticDSolveValue compute asymptotic approximations of solutions of odes for its independent variable approaching infinity? Thank to @Michael E2's comment, I found some examples in the document that consider this sort of limit. (Answer: yes, but for some problems)

  2. I am wondering what kind of expansion/approximation has been used in AsymptoticDSolveValue when it computes asymptotic approximations as $x\to\infty$. I ask this question because a common power series expansion $$f(x)\sim\sum_{n=0}^\infty c_n x^n$$ usually is appropriate for small $x$.

  3. If AsymptoticDSolveValue cannot solve this problem, is it possible to solve it with Series?

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    $\begingroup$ The answer to 1 is yes, if you use AsymptoticDSolveValue, but of course it does not solve every problem. $\endgroup$
    – Michael E2
    May 18, 2022 at 2:59
  • $\begingroup$ @MichaelE2 thanks a lot, I should use AsymptoticDSolveValue instead! I also updated the post. $\endgroup$
    – user95273
    May 18, 2022 at 3:27
  • $\begingroup$ The answer to 2 is, it depends on the given problem! For example AsymptoticDSolveValue[y''[x] - x y[x] == 0, y[x], {x, \[Infinity], 1} ] $c_1 e^{-\frac{2 x^{3/2}}{3}} \left(\frac{1}{\sqrt[4]{x}}-\frac{5}{48 x^{7/4}}\right)+c_2 e^{\frac{2 x^{3/2}}{3}} \left(\frac{5}{48 x^{7/4}}+\frac{1}{\sqrt[4]{x}}\right)$ $\endgroup$
    – Ulrich Neumann
    May 18, 2022 at 8:08
  • $\begingroup$ @UlrichNeumann, thank you. A power series expansion is usually not appropriate for $x\to\infty$. Is there any method to specify an appropriate kind of expansion in AsymptoticDSolveValue for a given problem? Or, could you suggest which kind of expansion is suitable for this equation? $\endgroup$
    – user95273
    May 18, 2022 at 11:49
  • $\begingroup$ The specification {x, ∞, n} is effectively a request to use an asymptotic series in the solution. $\endgroup$
    – J. M.'s eventual burnout ♦
    May 18, 2022 at 16:20

1 Answer 1

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All functions c1 x^aa with aa < 3/4 can serve as asymptotic solutions for deq. And combinations of them.

eq = D[x*f[x]^3*D[D[x*f'[x], x]/x, x], x];

ee[aa_] = eq /. f -> (c1 #^aa &) // Simplify

(*   2 aa^2 (2 - 5 aa + 2 aa^2) c1^4 x^(-3 + 4 aa)   *)

Equation tends to zero at infinity, if

Reduce[-3 + 4 aa < 0, aa]

(*   aa < 3/4   *)

Limit[ee[aa], x -> Infinity, Assumptions -> aa < (3/4)]

(*   0   *)

EDIT There are three overall solutions {c1, c1 Sqrt[x], c1 x^2} 

Solve[2 aa^2 (2 - 5 aa + 2 aa^2) == 0, aa]

(*   {{aa -> 0}, {aa -> 0}, {aa -> 1/2}, {aa -> 2}}   *)

eq /. f -> (c1 #^(2) &) // Simplify

(*   0   *)
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