Question

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

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

Answer 1

Answer: (C)

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