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How to solve the following equation:

$x \arcsin(x^2) + 2\sqrt{1-x^2}\arcsin(x) = 2x $

I think it is hard. Thanks.

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    You can use [Newton's method](http://en.wikipedia.org/wiki/Newton's_method)2012-10-12
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    $x=0$ $ { } $ $ { } $ $ { } $2012-10-12
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    @draks $x=0$ is only one of the solutions. There are more.2012-10-12
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    I checked that this has 3 roots. $x=0$ is easy to see, how about 2 other ones?2012-10-12

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There are only three roots, in $\{-u,0,u\}$, with $u$ pretty close to $1$. By convexity, the Newton method with starting point $x_0=\pm 1$ has quadratic convergence.

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    How can you get this has only three roots?2012-10-12