Question

if `y = e^(mcos^(-1)x)`, prove that `(1 - x^2) (d^2y)/(dx^2) - x dy/dx = m^2y`

Answer 1

`y = e^(mcos^(-1)x)`

`dy/dx = e^(mcos^(-1)) xx - m/((1-x^2))`

`(1-x^2)(dy/dx)^2 = m^2y^2`

`(1 - x^2).2 dy/dx (d^2y)/(ax^2) + (dy/dx)^2 (-2x) = m^22y dy/dx`

Dividing  `2 dy/dx`

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

hence proved