Question

The solution of the differential equation $x\left(y^2{e}^{xy}+e^{x/y}\right)dy=y\left(e^{x/y}-y^2{e}^{xy}\right)dx$ is

• A. $xy=\ln \left(e^{y/x}+c\right)$
• B. $xy=\ln \left(e^{x/y}+c\right)$
• C. $\frac{y}{x}=\ln \left(e^{xy}+c\right)$
• D. $\frac{x}{y}=\ln \left(e^{xy}+c\right)$

Answer 1

Answer: (B)

The given equation is
$x{y}^2{e}^{xy}dy+x{e}^{x/y}dy=y{e}^{x/y}dx-y^3{e}^{xy}dx$
$\Rightarrow y^2\cdot e^{xy}(xdy+ydx)=e^{x/y}(ydx-xdy)$
$\Rightarrow e^{xy}(xdy+ydx)=e^{x/y}\left(\frac{ydx-xdy}{y^2}\right)$
$\Rightarrow e^{xy}\cdot d(xy)=e^{x/y}(d(x/y))$
$\Rightarrow d\left(e^{xy}\right)=d\left(e^{x/y}\right)$
$\Rightarrow e^{xy}=e^{x/y}+c$
or $xy=\ln \left(e^{x/y}+c\right)$