Question

The general solution of the differential equation $(1+e^{\frac{x}{y}})dx+(1-\frac{x}{y})e^{\frac{x}{y}}dy=0$ is ($c$ is an arbitary constant)

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

Answer 1

The correct option is C $x+y{e}^{\frac{x}{y}}=c$

$dx+e^{\frac{x}{y}}dx+e^{\frac{x}{y}}dy-\frac{x}{y}{e}^{\frac{x}{y}}dy=0$
$\Rightarrow dx+e^{\frac{x}{y}}dy+e^{\frac{x}{y}}\frac{(ydx-xdy)}{y}=0$
$\Rightarrow dx+e^{\frac{x}{y}}dy+yd\left(e^{\frac{x}{y}}\right)=0$
$\Rightarrow dx+d\left(y{e}^{\frac{x}{y}}\right)=0$
Integrating both sides
$\Rightarrow x+y{e}^{\frac{x}{y}}=c$