Question

Solve $\frac{dy}{dx}=\frac{y^2}{xy-x^2}$

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

Answer 1

Answer: (C)

Given equation is $\frac{dy}{dx}=\frac{y^2}{xy-x^2}$ It is a homogeneous differential equation
Put $y=vx$ $\Rightarrow$ $\frac{dy}{dx}=v+x\frac{dv}{dx}$
$\therefore$ $v+x\frac{dv}{dx}=\frac{v^2{x}^2}{x\cdot vx-x^2}$
$\Rightarrow$ $v+x\frac{dv}{dx}=\frac{v^2}{v-1}$
$\Rightarrow$ $x\frac{dv}{dx}=\frac{v}{v-1}$
$\Rightarrow$ $\left(1-\frac{1}{v}\right)dv=\frac{dx}{x}$
On integrating, we get $v-\log v=\log x-\log c$
$\Rightarrow$ $\frac{y}{x}=\log \frac{y}{x}\cdot x\cdot \frac{1}{c}$
$\Rightarrow$ $y=c{e}^{y/x}$