Question

If $\frac{dy}{dx}=\frac{y}{x-\sqrt{xy}}$, then

• A. $\sqrt{\frac{x}{y}}=ln\,cy$
• B. $-\sqrt{\frac{x}{y}}=ln\,cy$
• C. $-2 \sqrt{\frac{x}{y}}=ln\,cy$
• D. $2 \sqrt{\frac{x}{y}}=ln\,cy$

Answer 1

Answer: (C)

Substitute Substitute $y=vx \Rightarrow \frac{dy}{dx}=\frac{xdv}{dx}+v$
Now, given equation becomes
$\frac{xdv}{dx}+v=\frac{v}{1-\sqrt{v}}$
$\Rightarrow \frac{xdv}{dx}=\frac{v}{1-\sqrt{v}}-v=\frac{v^{\frac{3}{2}}}{1-\sqrt{v}}$
$\Rightarrow \int \frac{1-\sqrt{v}}{v^{\frac{3}{2}}}dv=\int \frac{dx}{x}$
$\Rightarrow -2v ^{- \frac{1}{2}}-ln\,v=ln\,x+ln\,c$
$\Rightarrow -2 \sqrt{\frac{x}{y}}=ln\,cy$.