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Note: Some of the comments below (about 9) refer to an older version of this thread. I updated it to refer to the DSolve/NDSolve question.

Also: See All solutions to Lane-Edmon with n=5 for a complete set of general solutions to Lane-Edmon.

Consider the Lane-Edmon singular boundary-value problem: $$ y''+\frac{2}{t} y'+y^5=0,\quad y(1)=\sqrt{3/4},\quad y'(0)=0 $$ There are two solutions: $y_1(t)=\sqrt{\frac{3}{3+t^2}}$, $y_2=-\sqrt{\frac{3}{3+t^2}}$ although DSolve can not obtain these particular solutions it can and does return two extremely-complicated (implicit) general solutions, a part of one is shown below:

eqn1 = y''[t] + 2/t y'[t] + y[t]^5 == 0;
dSol = DSolve[eqn1, y[t], t]; (* takes about a minute *)

Solve[(2 Sqrt[3] Sqrt[t]
    EllipticF[
    ArcSin[Sqrt[((-Root[-12 C[1] - 3 t #1 + 4 t^3 #1^3 &, 1] + 
        Root[-12 C[1] - 3 t #1 + 4 t^3 #1^3 &, 3]) y[t]^2)/(
     Root[-12 C[1] - 3 t #1 + 4 t^3 #1^3 &, 
       3] (-Root[-12 C[1] - 3 t #1 + 4 t^3 #1^3 &, 1] + y[t]^2))]] . . .

For simpler expressions, we can fit an NDSolve solution to the implicit solution to confirm the accuracy of both. For example this thread: Implicit DSolve solution. And another thread: Implicit integral solution

Is it possible to do so in this case, that it, fit the two solutions returned by DSolve above by suitable values of the constants, to $y_1$ and $y_2$ and perhaps identify which of the seven families of solutions to Lane-Edmon the DSolve results belong to (see reference above)?

josh
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    Well, but y''[t] + 2/t y'[t] == y[t]^5 /. y -> sol1 // Simplify doesn't return True? – xzczd Feb 25 '22 at 14:51
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    This $y_1(t)=\sqrt{\frac{3}{3+t^2}}$ is not a solution. The authors Singh and Kumar made an error in the equation copying from another author who also made an error copying from another author who wrote it correctly :y''[t]+2/t y'[t]+y[t]^5==0 and then the mentioned function is indeed a solution of the problem at hand. By th way $y_2(t)=\sqrt{\frac{3 i}{3+t^2}}$, where $i=\sqrt{-1}$ might be a solution to the given equation. – Artes Feb 25 '22 at 14:58
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    NDSolve gives a correct answer. Try: Plot[{Evaluate@ NDSolveValue[{y''[t] + 2/t y'[t] == -y[t]^5, y[1] == Sqrt[3/4], y'[10^-15] == 0}, y[t], {t, 10^-15, 1}], Sqrt[3/(3 + t^2)]}, {t, 10^-15, 1}, PlotStyle -> {Red, {Black, Dashed}}]. See example1: https://springerplus.springeropen.com/articles/10.1186/s40064-016-2753-9 – Mariusz Iwaniuk Feb 25 '22 at 16:03
  • Ok. Thanks guys. Sorry, I should have confirmed the proposed solution satisfied the DE before posting the question. Also $y_2=-\sqrt{3/(3+t^2)]}$ is a second solution. – josh Feb 25 '22 at 16:34
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    Sorry, I'm a little confused: Is there still a question you want answered? (That is, it seems to be a mistake on the part of the authors of the paper, and that's all it is, right?) – Michael E2 Feb 25 '22 at 16:45
  • @josh I guess that this question stemmed from this related thread, right? https://mathematica.stackexchange.com/questions/264166/i-have-tried-two-different-techniques-to-find-the-exact-solution-of-bvp-but-cant?noredirect=1#comment658136_264166 –  Feb 25 '22 at 17:01
  • In any case, I think that we can all agree that there was an honest typo in the published paper -not the first time that this happened, definitely not the last one- and we can all verify the proposed solution, after the typo is fixed. I think that there is no ambiguity in comparing the numerical solution that Mma can offer to the analytic solution obtained by the authors. –  Feb 25 '22 at 17:03
  • Question to all: should we report this case to be included in the versions to come? –  Feb 25 '22 at 17:04
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    @Michael: Now that you mentioned it, yes: I've in the past found very worthwhile (education wise), "fitting" the implicit solutions returned by DSolve to the particular solutions returned by NDSolve. Is it possible to prove/show that $y_1$ and $y_2$ satisfy the the two general solutions returned by DSolve by substitution and appropriate values of the arbitrary constants? However, in this case, the implicit solutions are extremely complex so may not be possible. – josh Feb 25 '22 at 17:06
  • Okay, thanks. To confirm: you'd like this DSolve/NDSolve correspondence for your eqn1 = y''[t] + 2/t y'[t] == y[t]^5, not the "corrected" equation y''[t]+2/t y'[t]+y[t]^5==0 in the comments, right? – Michael E2 Feb 25 '22 at 17:24
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    @Michael: No. Let's keep the "corrected equation": $y''+2/ t y'+y^5=0$. A DSolve of that equation produces two extremely-complicated implicit integral equations. I'll look for a simpler one I worked on and post the link as an example of what I'm talking about. – josh Feb 25 '22 at 18:01
  • Then please edit your question so that is clear. We don't want the correct question buried in the comments. (Nobody has answered the question yet, so you could probably remove the irrelevant stuff or move it out of the way, like to the end or something. You could temporarily delete the question, if you need time to fix it.) – Michael E2 Feb 25 '22 at 18:03
  • @Michael: Done. – josh Feb 25 '22 at 18:15
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    When I was working on the other post I found the typo but completely missed this: it may be due to my lack of knowledge in complex analysis, but can a DE have a Boundary Condition at a point where it is undefined in real space and have real solutions? When we use these real solutions and multiply by $\frac{2}{t}$ are we not assuming $t\neq 0$? – MathX Feb 25 '22 at 22:51
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    @MathX See Mariusz's link. An elementary way to think of it is that if there is a solution $y$ such that the limit of the term, $\lim_{t\rightarrow0} y'(t)/t$, exists, then it may be possible to have a nonsingular solution at a singular point of the ODE. Consider this second-order version of the Power Rule: DSolve[{y''[t] == (n - 1) y'[t]/t, y[0] == 0, (*y[1] == 1*)}, y[t], t, Assumptions -> n > 1]. Does not always happen, though. – Michael E2 Feb 27 '22 at 19:50
  • @MichaelE2 I do remember it was mostly the limits that were allowed and not an actual function evaluation. Unfortunately the link is on a group of methods that I wasted several months of my PhD on. DTM, ADM, VIM, etc have thousands of publication on them but when I tried using them in my real world case they turned out to be mostly theoretical and mainly some extended form of Taylor expansion and they lose their accuracy away from the point of expansion. Also, many of the papers I saw had tried finding a solution and then evaluating a BC that worked for that solution which is not very helpful – MathX Feb 27 '22 at 21:11
  • @MathX I just meant that the paper discusses some conditions for the existence of a solution in the introductory "Background" section, since you raised a point about the singularity at the BC. – Michael E2 Feb 28 '22 at 01:09
  • @michaelE2 ahh I see thanks. – MathX Feb 28 '22 at 04:03

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