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How can I show that the elliptic integral of the second kind

$E(x) = \int_0^{\pi/2} \sqrt{1-x^2\sin^2(t)} \,dt $

satifies the equation

$$E''(x)\, (x^2 -1)x + E'(x) \,(x^2-1) - E(x) = 0.$$

where $E'' , E'$ represent the first and second derivatives respectively?

Edit: I tried using the Legendre relation, but so far I am still very stuck....

Adrian Keister
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King
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  • Your "equation" involving $E'', E',$ and $E$ is not actually an equation - I don't see an equals sign anywhere. Can you please edit to make it an equation? You also need to show what work you've done on the problem, or this question will likely be closed due to lack of context. – Adrian Keister Mar 29 '19 at 14:56
  • @AdrianKeisterThanks for the help...This is my first time here.... – King Mar 29 '19 at 15:04
  • Right, that's why I'm helping you learn M.SE culture. M.SE rewards questions that are well-documented and show effort on the asker's part, and punishes questions that are of the type "Please do this for me." Your edit is still not sufficient. Please type up your calculations. – Adrian Keister Mar 29 '19 at 15:21
  • See https://math.stackexchange.com/questions/426257 and https://math.stackexchange.com/questions/198480 – cgiovanardi Mar 30 '19 at 00:36
  • Use the fact that $E(x) ={} _2F_1(-1/2,1/2;1;x^2)$ and use the differential equation for hypergeometric function. – Paramanand Singh Mar 30 '19 at 05:35
  • For the hypergeometric differential equation see this post. – Paramanand Singh Mar 30 '19 at 05:40

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