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

y=(sin^-1 x)²

sin^-1x= y^1/2 ..........1

 x=sin(y)^1/2

differentiating wrt x

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

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

again differentiating wrt x

y'' = 2[ (y)^1/2 secy^1/2 tany^1/2 (1/2y^1/2)  + secy^1/2 (1/2y^1/2) ]  .....................3

multiplying 1 and 2

xy' =2y^1/2 tany^1/2 ...............4

solving these equations we wil get the desired result