Question

The solution of differential equation $(x^2-xy)dy=(xy+y^2)dx$ is :

• A. $xy=c{e}^{-\frac{y}{x}}$
• B. $xy=c{e}^{-\frac{x}{y}}$
• C. $y{x}^2=c{e}^{\frac{1}{x}}$
• D. None of these

Answer 1

The correct option is B $xy=c{e}^{-\frac{x}{y}}$

Given differential eqn is

$(x^2-xy)dy=(xy+y^2)dx$

$\Rightarrow \frac{dy}{dx}=\frac{xy+y^2}{x^2-xy}$ ....(1)

which is a homogeneous differential eqn

Put $y=vx$

$\frac{dy}{dx}=v+x\frac{dv}{dx}$

So, eqn (1) becomes

$v+x\frac{dv}{dx}=\frac{v+v^2}{1-v}$

$\Rightarrow x\frac{dv}{dx}=\frac{2v^2}{1-v}$

$\Rightarrow \frac{1-v}{2v^2}dv=\frac{dx}{x}$

Integrating both sides, we get

$\Rightarrow \int \frac{1-v}{v^2}dv=2\int \frac{dx}{x}$

$\Rightarrow \frac{-1}{v}-\log v=2\log x+\log C$

$\Rightarrow \frac{-x}{y}-\log y+\log x=2\log x+\log C$

$\Rightarrow \frac{-x}{y}=\log xyC$

$\Rightarrow c{e}^{-x/y}=xy$ where $c=\frac{1}{C}$