Form the differential equation of the family curves having equation `y=(sin^(-1)x)^(2)+Acos^(-1)x+B`.

Explanation

`y=(sin^(-1)x)^(2)+Acos^(-1)x+B`
`=(sin^(-1)x)^(2)-Asin^(-1)sin^(-1)x+(piA)/2+B`
Differentiating w.r.t.x, we get
`(dy)/(dx) = (2sin^(-1)x)/sqrt(1-x^(2))-A/sqrt(1-x^(2))`
`rArr (1-x^(2))((dy)/(dx))^(2)=4(sin^(-1)x)^(2)-4Asin^(-1)x+A^(2)`
`=4y-2piA-4B+A^(2)`
Differentiating again, w.r.t, x we get
`2(1-x^(2))(dy)/(dx)(d^(2)y)/(dx^(2))-2x((dy)/(dx))^(2)=4(dy)/(dx)`
`rArr (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)=2`, which is required differential equation.