let-give-J-x-1-pi-0-pi-cos-xcost-dt-1-find-J-and-J-in-form-of-integrals-2-prove-that-J-x-x-pi-0-pi-sin-2-t-cos-xcost-dt-and-J-is-solution-of-d-e-xy-y-xy-0-

Tinku Tara June 4, 2023 Integration 0 Comments

Question Number 30215 by abdo imad last updated on 18/Feb/18

let give J(x)= (1/π) ∫_0 ^π cos(xcost)dt 1) find J^′ and J^(′′) in form of integrals 2)prove that J^′ (x)=((−x)/π) ∫_0 ^π sin^2 t cos(xcost)dt and J is solution of d.e. xy^(′′) +y^′ +xy=0

$${let}\:{give}\:{J}\left({x}\right)=\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {cos}\left({xcost}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{J}^{'} \:{and}\:{J}^{''} \:{in}\:{form}\:{of}\:{integrals} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{J}^{'} \left({x}\right)=\frac{−{x}}{\pi}\:\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{2}} {t}\:{cos}\left({xcost}\right){dt}\:{and}\:{J}\:{is} \\ $$$${solution}\:{of}\:{d}.{e}.\:\:{xy}^{''} \:+{y}^{'} \:+{xy}=\mathrm{0} \\ $$

Facebook Tweet Pin

let-give-J-x-1-pi-0-pi-cos-xcost-dt-1-find-J-and-J-in-form-of-integrals-2-prove-that-J-x-x-pi-0-pi-sin-2-t-cos-xcost-dt-and-J-is-solution-of-d-e-xy-y-xy-0-

Leave a Reply Cancel reply