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if y =(arcsin x)^2 ,prove that (1-x^2)y" - xy' - y=0 where y" is second derivative of y with respect to x y' is first derivative of y with respect to x

if y =(arcsin x)^2 ,prove that (1-x^2)y" - xy' - y=0


where y" is second derivative of y with respect to x


        y' is first derivative of y with respect to x

Grade:10

1 Answers

vikas askiitian expert
509 Points
10 years ago

y=(sin-1 x)2

sin-1x= y1/2 ..........1

 x=sin(y)1/2

differentiating wrt x

1 = cos(y)1/2 y' /2y1/2 or

 y'=2 (y)1/2 sec(y)1/2  ........................2

again differentiating wrt x

y'' = 2[ (y)1/2 secy1/2 tany1/2 (1/2y1/2)  + secy1/2 (1/2y1/2) ]  .....................3

multiplying 1 and 2

xy' =2y1/2 tany1/2 ...............4

solving these equations we wil get the desired result

 

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