`"If "y=e^(a cos^(-1)x),-1lexle1," show that "(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)-a^(2)y=0.`

Explanation

`y=e^(acos^(-1)x)`
`implies(dy)/(dx)=e^(acos^(-1)x)*(d)/(dx)(acos^(-1)x)`
`implies (dy)/(dx)=y*((-a))/(sqrt(1-x^(2)))`
`impliessqrt(1-x^(2))(dy)/(dx)= -ay`
`implies(1-x^(2))((dy)/(dx))^(2)=a^(2)y^(2)`
Differentiate both sides w.r.t.x
`(1-x^(2))*2(dy)/(dx)*(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)(-2x)=a^(2)*2y(dy)/(dx)`
Divide both sides by `2*(dy)/(dx)`
`(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)=a^(2)y`
`(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)-a^(2)y=0 " " ` Hence Proved.