0
$\begingroup$

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?

$\endgroup$
5
  • 2
    $\begingroup$ The answer to 1 is yes, if you use AsymptoticDSolveValue, but of course it does not solve every problem. $\endgroup$
    – Michael E2
    May 18 at 2:59
  • $\begingroup$ @MichaelE2 thanks a lot, I should use AsymptoticDSolveValue instead! I also updated the post. $\endgroup$
    – user95273
    May 18 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} ] $c_1 e^{-\frac{2 x^{3/2}}{3}} \left(\frac{1}{\sqrt[4]{x}}-\frac{5}{48 x^{7/4}}\right)+c_2 e^{\frac{2 x^{3/2}}{3}} \left(\frac{5}{48 x^{7/4}}+\frac{1}{\sqrt[4]{x}}\right)$ $\endgroup$ May 18 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 at 11:49
  • $\begingroup$ The specification {x, ∞, n} is effectively a request to use an asymptotic series in the solution. $\endgroup$ May 18 at 16:20

1 Answer 1

1
$\begingroup$

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   *)
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.