If `y=e^acos^((-1)x),-1lt=x<1,s howt h a t` `(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)-a^2y=0`

Explanation

`y=e^(a cos^(-1)x)`
`therefore" "(dy)/(dx)=e^(a cos^(-1)x)(-a)/(sqrt(1-x^(2)))=(-ay)/(sqrt(1-x^(2)))`
`"or "(1-x^(2))((dy)/(dx))^(2)=a^(2)y^(2)`
Differentiating both sides with respect to x, we get
`((dy)/(dx))^(2)(-2x)+(1-x^(2))xx2(dy)/(dx)cdot(d^(2)y)/(dx^(2))=a^(2)cdot2ycdot(dy)/(dx)`
`"or "-x(dy)/(dx)+(1-x^(2))(d^(2)y)/(dx^(2))=a^(2).y[(dy)/(dx)ne0]`
`"or "(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)-a^(2)y=0`