Question

The solution of the differential equation
$ \frac{dy}{dx} = \frac {2xy - y^2}{2xy - x^2}, $ is

• A. $xy(x+ y) = C$
• B. $xy(x-y) = C$
• C. $ x^2y(x-y) = C $
• D. $ x^3y(x-y) = C $

Answer 1

Answer: (B)

We have,
$\frac{d y}{d x} =\frac{2 x y-y^{2}}{2 x y-x^{2}} .... $ (i)
Let $y =v x ....$ (ii)
$\Rightarrow \frac{d y}{d x} =v+x \frac{d v}{d x} ....$ (iii)
On putting the values from Eqs. (ii) and (iii) in Eq. (i), we get
$v+x \frac{d v}{d x}=\frac{2 v-v^{2}}{2 v-1}$
$\Rightarrow x \frac{d v}{d x}=\frac{2 v-v^{2}}{2 v-1}-v$
$\Rightarrow x \frac{d v}{d x}=\frac{2 v-v^{2}-2 v^{2}+v}{2 v-1}$
$\Rightarrow x \frac{d v}{d x}=\frac{3 v-3 v^{2}}{2 v-1} $
$\Rightarrow \frac{2 v-1}{v^{2}-v} d v =\frac{-3}{x} d x$
$\Rightarrow \int\left(\frac{2 v-1}{v^{2}-v}\right) d v=-3 \int \frac{d x}{x}$
$\Rightarrow \log \left|v^{2}-v\right|=-3 \log x+\log c$
$\Rightarrow \log \left(v^{2}-v\right) x^{3}=\log c$
$\Rightarrow \left(\frac{y^{2}}{x^{2}}-\frac{y}{x}\right) x^{3}=c$
$\Rightarrow x\left(y^{2}-x y\right) \equiv c$
$\Rightarrow x y(y-x)=c$
$\Rightarrow x y(x-y)=c$