Question

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

Answer 1

We have y = `e^(a cos^(-1)x)`  ...(1)

Differentiating (1) both sides w.r.t. x, we get

`dy/dx = e^(a cos^(-1)x) d/dx (a cos^-1 x)`

`= e^(a cos^(-1)x) ((- a)/sqrt(1 - x^2))`

`= (- ay)/(sqrt(1 - x^2))`   ...(2)

Differentiating (2) both sides w.r.t. x, we get

`(d^2y)/(dx^2) = -a[(sqrt(1-x^2) dy/dx - y d/dx sqrt(1 - x^2))/((1-x^2))]`

`(d^2y)/(dx^2) = -a[(sqrt(1-x^2)dy/dx - y/(2sqrt(1-x^2)) * (-2x))/((1-x^2))]`

`(1 - x^2) (d^2y)/dx^2 = -a[-ay + (xy)/sqrt(1-x^2)]` ....[from (2)]

`(1 - x^2) (d^2y)/dx^2 = -a[-ay + x * ((-1)/a * dy/dx)]`

`(1 - x^2) (d^2y)/(dx^2) = a^2y + x dy/dx`

`(1 - x^2) (d^2y)/(dx^2) - x dy/dx - a^2y = 0`