Question

The general solution of the differential equation $x(y^2{e}^{xy}+e^{x/y})dy=y(e^{x/y}-y^2{e}^{xy})dx$ is
(Where $c$ is the constant of integration)

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

Answer 1

The correct option is D $e^{xy}=e^{x/y}+c$

$x(y^2{e}^{xy}+e^{x/y})dy=y(e^{x/y}-y^2{e}^{xy})dx$
$\Rightarrow x{y}^2{e}^{xy}dy+x{e}^{x/y}dy=y{e}^{x/y}dx-y^3{e}^{xy}dx$
$\Rightarrow x{y}^2{e}^{xy}dy+y^3{e}^{xy}dx=y{e}^{x/y}dx-x{e}^{x/y}dy$
$\Rightarrow y^2{e}^{xy}(xdy+ydx)=e^{x/y}(ydx-xdy)$
Divide by $y^2$, we get
$e^{xy}(xdy+ydx)=e^{x/y}\left(\frac{ydx-xdy}{y^2}\right)$
$\Rightarrow e^{xy}\left(d\right(xy\left)\right)=e^{x/y}\left(d\left(x/y\right)\right)$
$\Rightarrow \int d\left(e^{xy}\right)=\int d\left(e^{x/y}\right)$
$\therefore e^{xy}=e^{x/y}+c$