If x=sint and y=cos pt, then

Options

A. `(1-x^(2))y_(2)+xy_(1)+p^(2)y=0`
B. `(1-x^(2))y_(2)+xy_(1)-p^(2)y=0`
C. `(1+x^(2))y_(2)-xy_(1)+p^(2)y=0`
D. `(1-x^(2))y_(2)-xy_(1)+p^(2)y=0`

Answer

D

Explanation

Given x=sin t, y=cos pt
`(dx)/(dt)=cost`
`(dy)/(dt)=-psinpt`
`therefore(dy)/(dx)=-(psinpt)/(cost)`
`impliesy_(1)=(-psqrt(1-y^(2)))/(sqrt(1-x^(2)))`
`impliesy_(1)sqrt(1-x^(2))=-psqrt(1-y^(2))`
`impliesy_(1)^(2)(1-x^(2))=p^(2)(1-y^(2))`
`implies2y_(1)y_(2)(1-x^(2))-2xy_(1)^(2)=-2y y_(1)p^(2)` [differentiating]
`implies(1-x^(2))y_(2)-xy_(1)+p^(2)y=0`