If `y=(1/2)^(n-1)cos(ncos^(-1)x),` then prove that `y` satisfies the differential equation `(1-x^2)` `(d^2y)/(dx^2)-x(dy)/(dx)+n^2y=0`

Explanation

`y=(1//2^(n-1))cos (n cos^(-1)x)`
`therefore" "(dy)/(dx)=-(1)/(2^(n-1))sin(n cos^(-1)x)[(-n)/sqrt((1-x^(2)))]`
`"or "(1-x^(2))((dy)/(dx))^(2)=(n^(2))/(2^(2n-2))sin^(2)(n cos^(-1)x)`
`=(n^(2))/(2^(2n-2))[1-cos^(2)(n cos^(-1)x)]`
`=n^(2)[(1)/(2^(2n-2))-y^(2)]`
Differentiating both sides w.r.t x, we get
`(1-x^(2))2(dy)/(dx)(d^(2)y)/(dx^(2))-2x((dy)/dx)^(2)=-2n^(2)y(dy)/(dx)`
`"or "(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)+n^(2)y=0`