g-is-real-function-continue-let-f-x-0-x-sin-x-t-g-t-dt-1-prove-that-f-x-0-x-cos-t-x-g-t-dt-2-prove-that-f-is-so-lt-ution-of-the-diff-equa-y-y-g-x-

Tinku Tara June 4, 2023 Integration 0 Comments

Question Number 31507 by abdo imad last updated on 09/Mar/18

g is real function continue let f(x)=∫_0 ^x sin(x−t)g(t)dt 1)prove that f^′ (x)= ∫_0 ^x cos(t−x)g(t)dt 2)prove that f is so<ution of the diff.equa. y^(′′) +y =g(x)

$${g}\:{is}\:{real}\:{function}\:{continue}\:{let} \\ $$$${f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{sin}\left({x}−{t}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}^{'} \left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} {cos}\left({t}−{x}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{so}<{ution}\:{of}\:{the}\:{diff}.{equa}. \\ $$$${y}^{''} \:+{y}\:={g}\left({x}\right) \\ $$

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