Question

Solve the differential equation $x(x-y)dy=y(x+y)dx$.

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

Answer 1

The correct option is C $kxy=e^{-x/y}$

$x(x-y)dy=y(x+y)dx\Rightarrow x(x-y)\frac{dy}{dx}=y(x+y)$
Substitute $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$

$\therefore x(x-xv)(x\frac{dv}{dx}+v)=x^2(v+1)v$

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

Integrating both sides we get
$\int \frac{v-1}{v^2}\frac{dv}{dx}dx=-\int \frac{2}{x}dx$
$\Rightarrow \log v+\frac{1}{v}=-2\log x+c\Rightarrow kxy=e^{-x/y}$