let g(x)=(√(−x+(√(1+x^2 )))) 1) prove that g is solution for the differencial equation 4(1+x^2 )y^(′′) +4xy^′ −y =0 .prove that g is C^∞ on R 2) determine a relation between g^((n)) (0) and g^((n+2)) (0)


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Question Number 40103 by maxmathsup by imad last updated on 15/Jul/18

$l e t g (x) = \sqrt{- x + \sqrt{1 + x^{2}}}$ $1) p r o v e t h a t g i s s o l u t i o n f o r t h e d i f f e r e n c i a l e q u a t i o n$ $4 (1 + x^{2}) y^{^{″}} + 4 {x y}^{^{'}} - y = 0 . p r o v e t h a t g i s C^{\infty} o n R$ $2) d e t e r m i n e a r e l a t i o n b e t w e e n g^{(n)} (0) a n d g^{(n + 2)} (0)$
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