Question

The solution of $x^2 y_1^2+x y y_1-6 y^2=0$ are

• A. $y=C x^2$
• B. $x^3 y=C$
• C. $\frac{1}{2} \ln y=C+\ell n x$
• D. All of these

Answer 1

Answer: (D)

$x^2 y_1^2+x y y_1-6 y^2=0$
It is quadratic equation in $y _1$
$ y_1=\frac{-x y \pm \sqrt{x^2 y^2+24 y^2 x^2}}{2 x^2}=\frac{-x y \pm 5 x y}{2 x^2}$
$y_1=-\frac{3 y}{x} \mid y_1=\frac{2 y}{x} $
$ \frac{d y}{d x}=\frac{-3 y}{x} \mid \frac{d y}{d x}=\frac{2 y}{x} $
$ -\frac{ dy }{ y }=3 \frac{ dx }{ x } | \frac{ dy }{ dx }=\frac{2 y }{ x }$
$ -\ell \ln y=3 \ln x+\ln c | \ln y=2 \ln x+\ell nc $
$ x^3 y=C y=c x^2$
Option (3)
$ \ln y=c+2 \ell \ln x $
$ \frac{1}{2} \ln y=\ell n c_1+\ell n x$