Question

If $x\left(y-x\right)\frac{dy}{dx}=y\left(y+x\right)$, then

• A. $xy=c{e}^{y/x}$
• B. $x=cy{e}^{y/x}$
• C. $y=cx{e}^{y/x}$
• D. $xy{e}^{y/x}=c$

Answer 1

Answer: (A)

$\frac{dx}{dy}$$=\frac{y\left(y+x\right)}{x\left(y-x\right)}$
Substitute $y=vx$
$\Rightarrow \frac{dy}{dx}=\frac{xdv}{dx}+v$
Now, given equation becomes
$\frac{xdv}{dx}+v=\frac{v\left(v+1\right)}{\left(v-1\right)}$
$\Rightarrow \frac{xdv}{dx}=\frac{v\left(v+1\right)}{v-1}-v=\frac{2v}{v-1}$
$\Rightarrow \frac{dx}{x}=\left(\frac{v-1}{2v}\right)dv$
$\Rightarrow 2lnx=v-lnv+lnc$
$\Rightarrow lnxy=\frac{y}{x}+lnc$
$\Rightarrow xy=c{e}^{\frac{y}{x}}$.