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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
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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