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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`
Solution
y = `e^(a cos^(-1)x)`
On differentiating with respect to x,
`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))`
On multiplying by `sqrt(1 - x^2)`,
`=> sqrt(1 - x^2) dy/dx` = - ay
On squaring,
`(dy/dx)^2 (1 - x^2) = a^2y^2`
Differentiating again with respect to x,
`=> 2 (dy/dx) (d^2y)/dx^2 (1 - x^2) + (dy/dx)^2 (- 2x) = 2a^2 y dy/dx`
Dividing by `2 dy/dx`,
`(d^2y)/dx^2 (1 - x^2) - x dy/dx = a^2 y`
Hence, `(1 - x^2) (d^2y)/dx^2 - x(dy/dx) - a^2 y = 0`
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