# Can Mathematica solve an ode asymptotically as x goes to infinity?

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?

• The answer to 1 is yes, if you use AsymptoticDSolveValue, but of course it does not solve every problem. Commented May 18, 2022 at 2:59
• @MichaelE2 thanks a lot, I should use AsymptoticDSolveValue instead! I also updated the post. Commented May 18, 2022 at 3:27
• 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} ] $c_1 e^{-\frac{2 x^{3/2}}{3}} \left(\frac{1}{\sqrt[4]{x}}-\frac{5}{48 x^{7/4}}\right)+c_2 e^{\frac{2 x^{3/2}}{3}} \left(\frac{5}{48 x^{7/4}}+\frac{1}{\sqrt[4]{x}}\right)$ Commented May 18, 2022 at 8:08
• @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? Commented May 18, 2022 at 11:49
• The specification {x, ∞, n} is effectively a request to use an asymptotic series in the solution. Commented May 18, 2022 at 16:20

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   *)