If a function is represented parametrically be the equations `x=(1+(log)_e t)/(t^2); y=(3+2(log)_e t)/t ,` then which of the following statements are true? `y^(x-2x y^(prime))=y` `y y^(prime)=2x(y^(prime))^2+1` `x y^(prime)=2y(y^(prime))^2+2` `y^(y-4x y^(prime))=(y^(prime))^2`

Options

A. `y''(x-2xy')=y`
B. `yy'=2x(y')^(2)+1`
C. `xy'=2y(y')^(2)+2`
D. `y''(y-4xy')=(y')^(2)`

Explanation

`x=(1+log_(e)t)/(t^(2)),y=(3+2 log_(e)t)/(t)`
`(dy)/(dx)=(dy//dt)/(dx//dt)`
`=((t((2)/(t))-(3+ 2 log_(e)t))/(t^(2)))/(t^(2)((1)/(t))-(1+log_(e)t)2t)`
`((-1-2log_(e)t)/(-1-2log_(e)t))t=t`
Eliminating `log_(e)t` term from y, we get
`y=(1+2t^(2)x)/(t)=(1+2(y')^(2)x)/(y')`
`"or "yy'=1+2x(y')^(2)" (Differentiating w.r.t. x)"`
`"or "yy''+(y')^(2)=4xy'cdoty''+2(y')^(2)`
`"or "yy''=4xycdoty''+(y')^(2)`