If `y=e^acos^((-1)x)` , `-1lt=xlt=1` , show that `(1-x^2)` `(d^2y)/(dx^2)-x(dy)/(dx)-a^2y=0` .

Explanation

Given curve is `y=e^{a cos ^{-1} x}`

Differentiating given curve,
`
Rightarrow y^{prime}=e^{a cos ^{-1} x} cdot frac{-a}{sqrt{(1-x^{2})}}`
`
Rightarrow y^{prime}=frac{-a y}{sqrt{(1-x^{2})}} ldots ldots . . .(1) `

On differentiating above equation again w.r.t x, we get
`
Rightarrow y^{prime prime}=frac{-a(-a e^{a cos ^{-1} x}+frac{x cdot e^{a cos ^{-1} x}}{sqrt{(1-x^{2})}})}{(1-x^{2})}`
` Rightarrow(1-x^{2}) y^{prime prime}=-a(-a e^{a cos ^{-1} x}+frac{x cdot e^{a} cos ^{-1} x}{sqrt{(1-x^{2})}}) ldots .( from equation (1)) . `
`Rightarrow(1-x^{2}) y=a^{2} e^{a cos ^{-1} x}+x cdot y) `
` Rightarrow(1-x^{2}) y=x cdot y-a^{2} cdot y=0 `
`Rightarrow(1-x^{2}) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}-a^{2} y=0 `
hence proved