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If y = e^(a cos^(-1)), - 1 le x le 1 ,...

If ` y = e^(a cos^(-1)), - 1 le x le 1 ,` the which of these are correct ?

A

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

B

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

C

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

D

None of the above

Text Solution

Verified by Experts

The correct Answer is:
C
Given , ` y = e^(a cos ^(-1)x)` … (i)
On differentiating both sides w.r.t.x, we get
and `(dy)/(dx) = e^(a cos ^(-1)x). (-a)/(sqrt(1 - x)^(2)) = (-ay)/(sqrt(1 -x^(2)))" "[ because e^(a cos ^(-1)x) = y]`
`rArr sqrt(1 -x^(2))(dy)/(dx) = - ay` ...(ii)
Again , differnetiating both sides w.r.t.x, we get
`sqrt(1 -x^(2))(d^(2)y)/(dx^(2) ) + (dy)/(dx) .(-2x)/(2sqrt(1 -x^(2))) = - a(dy)/(dx)`
`sqrt(1 -x^(2))(d^(2)y)/(dx^(2) ) +((-x)/(sqrt(1 -x^(2)))) (dy)/(dx) = (- a(-ay))/(sqrt(1 - x^(2)))" "` [using Eq. (ii)]
`rArr (1 + x^(2)) (d^(2) y)/(dx^(2)) - x (dy)/(dx) - a^(2) y = 0 ` .
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