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I wonder if there are more closed form (preferably, in elementary functions or basic special functions, like Zeta, Gamma and Polygamma) solutions for the differential equation

$f'(x)=B\left(\frac{1}{2},a\right) \sqrt{f(x)} (1-f(x))^{1-a},f(0)=0,$

at different $a$? Here $B$ is the Euler's Beta-function. So far, I found 3 solutions, all of them elementary functions ($S_a(x)$):

  • $S_{\frac{1}{2}}(x)=\sin ^2\left(\frac{\pi x}{2}\right)$
  • $S_1(x)=x^2$
  • $S_2(x)=\sqrt[3]{2 x^2+2 \sqrt{x^2 \left(x^2-1\right)}-1}+\frac{1}{\sqrt[3]{2 x^2+2 \sqrt{x^2 \left(x^2-1\right)}-1}}+2$

I am particularly interested in a closed form for $S_{3/2}(x)$, that is, for $a=3/2$.

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  • $\begingroup$ @MichaelEngelhardt are you sure? It looks like the graphic is not what expected. $\endgroup$
    – Anixx
    Commented Nov 10 at 15:28
  • $\begingroup$ @MichaelEngelhardt still, it does not look as what expected. I expect this graphic: i.imgur.com/fIgGOtM.png $\endgroup$
    – Anixx
    Commented Nov 10 at 15:46
  • $\begingroup$ @MichaelEngelhardt still, it has a very different plot... $\endgroup$
    – Anixx
    Commented Nov 10 at 15:58
  • $\begingroup$ @MichaelEngelhardt by the way, I've got the integral from the answer by Alexandre as $\sqrt{1-f} \sqrt{f}-\frac{1}{2} \sin ^{-1}(1-2 f)$, using Rubi. $\endgroup$
    – Anixx
    Commented Nov 10 at 16:05
  • $\begingroup$ Ok, $x=2\sqrt{f(1-f)}\ / \pi - (1/\pi ) \arctan(\sqrt{(1-f)/f}\ (2f-1)/(2f-2))+1/2$. $\endgroup$
    – Michael Engelhardt
    Commented Nov 10 at 16:10

1 Answer 1

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Your differential equation is separable, and is equivalent to integration of $$\int f^{-1/2}(1-f)^{a-1}df.$$ According to a theorem of Chebyshev, this integral is elementary if and only if either $a$ is an integer or $a-1/2$ is an integer.

This theorem is mentioned in Russian Calculus textbooks, for example, in G. M. Fichtenholz, G. M. Differential- und Integralrechnung. III. Translated from the Russian by Ludwig Boll and Konrad Gröger. Eleventh edition. There is no complete English translation. He explains how to integrate it in all cases when the integral is elementary.

The original paper of Chebyshev (in French) is available in the first volume of his collected works, pages 147-168. His collected works are available online, for example here

Remarks. Some parts of Fichtenholtz are translated into English, the relevant part seems to be:

MR0354950 (50 #7427) Fichtenholz, G. M. The indefinite integral. Translated and freely adapted from the Russian by Richard A. Silverman. The Pocket Mathematical Library, Course 6. Gordon and Breach Science Publishers, New York-London-Paris, 1971.

but I do not have an access to this translation.

  1. Another place to look for an explicit integral is various tables of integrals, the most comprehensive set is by Gradshteyn and Ryzhyk
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  • $\begingroup$ This is great! And what's the solution for $a=3/2$ in closed form? $\endgroup$
    – Anixx
    Commented Nov 10 at 15:13
  • $\begingroup$ I only get this integral as an implicit function. $\endgroup$
    – Anixx
    Commented Nov 10 at 15:16
  • $\begingroup$ It is not always possible to write it as an explicit elementary function. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 10 at 15:25

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