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. $x^{2}y(x-y)=C$
• D. $x^{3}y(x-y)=C$

Answer 1

Answer: (B)

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