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

To solve the problem, we need to differentiate the parametric equations given and check which of the statements are true. The equations are:

1. \( x = \frac{1 + \log_e t}{t^2} \)
2. \( y = \frac{3 + 2 \log_e t}{t} \)

### Step 1: Differentiate \( x \) with respect to \( t \)

Using the quotient rule, which states that if \( u \) and \( v \) are functions of \( t \), then:
...