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I get a useless answer for the problem below, symbolically, although find_root works in Sagemath, and if you graph it the approximate solution is easily visible for finding the root numerically. However, is there any more advanced method, such as infinite descent, to solving such equations non-numerically? I tried but couldn't do it.

solve(x^2-5==sin(x),x)

answer: [x == -sqrt(sin(x) + 5), x == sqrt(sin(x) + 5)]

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    $\begingroup$ no, the equation $x^2-5=\sin x$ does not have a closed-form solution (the numerical solution is $x=-2.02521$) $\endgroup$ – Carlo Beenakker yesterday
  • $\begingroup$ @Carlo Beenakker... or $x=2.3847...$ $\endgroup$ – Jean Marie Becker 43 mins ago

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