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Question Number 30749 by abdo imad last updated on 25/Feb/18

let f(x)=arcsinx with x∈[0,1]  1) prove that (1−x^2 )f^(′′) (x) −xf^′ (x)=0  2)prove that (1−x^2 )f^((n+2)) (x)=(2n+1)x f^((n+1)) (x) +n^2 f^((n)) (x)  3) prove that  f^((n)) (x) ≥0 ∀n .

$${let}\:{f}\left({x}\right)={arcsinx}\:{with}\:{x}\in\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{''} \left({x}\right)\:−{xf}^{'} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)=\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+{n}^{\mathrm{2}} {f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left({n}\right)} \left({x}\right)\:\geqslant\mathrm{0}\:\forall{n}\:. \\ $$

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