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


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

$l e t g i v e J (x) = \frac{1}{π} \int_{0}^{π} c o s (x c o s t) d t$ $1) f i n d J^{^{'}} a n d J^{^{″}} i n f o r m o f i n t e g r a l s$ $2) p r o v e t h a t J^{^{'}} (x) = \frac{- x}{π} \int_{0}^{π} {s i n}^{2} t c o s (x c o s t) d t a n d J i s$ $s o l u t i o n o f d . e . {x y}^{^{″}} + y^{^{'}} + x y = 0$
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