Question Number 30747 by abdo imad last updated on 25/Feb/18
$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{d}.{e}.{wich}\:{verify}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall{x}\in{R}\:,\forall{n}\in{N} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)+\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+\left({n}^{\mathrm{2}} −\mathrm{1}\right){f}^{\left({n}\right)} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left(\mathrm{2}{n}+\mathrm{1}\right)} \left(\mathrm{0}\right)=\mathrm{0}\:\forall{n}\in\:{N} \\ $$