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If x=sint and y=sinp t , prove that (1-x...

If `x=sint` and `y=sinp t ,` prove that `(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^2y=0.`

A

`-y`

B

y

C

py

D

`-p^(2)y`

Text Solution

Verified by Experts

The correct Answer is:
D
Given that , ` x = sin t " and " y = sin pt `
` rArr (dx)/(dt) = cos t " and " (dy)/(dt) = p * cos pt`
` therefore (dy)/(dx) = (p cos pt )/(cos t )` …(i)
` rArr (dy)/(dx) = (psqrt(1 - sin^(2) pt))/(sqrt( 1 - sin ^(2) t)) = p sqrt((1 - y^(2))/(1 - x^(2)))`
` rArr ((dy)/(dx))^(2) (1 - x^(2)) = p^(2) (1 - y^(2))`
On differentiating both sides w.r.t.w, we get
`(1 - x^(2) )2 (dy)/(dx).(d^(2) y)/(dx^(2)) + ((dy)/(dx))^(2) (-2x) = p^(2) (-2p) (dy)/(dx)`
or ` (1 - x^(2)) (d^(2) y)/( dx^(2)) - xx(dy)/(dx) = - p^(2) y`
[ dividing both sides by ` 2 (dy_(1))/(dx) ] `
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