Question

The solution of the differential equation $ \frac{dy}{dx} = xe^{x-y}$ is

• A. $e^{x-y} = c$
• B. $e^y (xe^x + e^x) + c + 1 = 0$
• C. $e^{x-y} = e^{xy}$
• D. none of these

Answer 1

Answer: (D)

$\frac{dy}{dx} =xe^{x-y}=xe^{x}.e^{-y}$
$ e^{y }dy = xe^{x} dx $
Integrating both sides , we get
$\int e^{y}dy = \int xe^{x }dx + c$
$\Rightarrow e^{y} =xe^{x} - \int1 . e^{x}dx + c $
$\Rightarrow e^{y} =xe^{x} -e^{x}+c$
$ \Rightarrow e^{y} -e^{x}\left(x-1\right)+c =0$