Question

The solution of the differential equation $\frac{dy}{dx}=e^{x-y}+x^{2}\,e^{-y}$ is

• A. $y=e^{x-y}-x^{2}\,e^{-y}+c$
• B. $e^{y}-e^{x}=\frac{x^{3}}{3}+c$
• C. $e^{y}+e^{y}=\frac{x^{3}}{3}+c$
• D. $e^{y}-e^{y}=\frac{x^{3}}{3}+c$

Answer 1

Answer: (B)

$\frac{dy}{dx}=e^{x-y}+x^{2}\,e^{-y}$
$\Rightarrow \frac{dy}{dx}=e^{-y}\left(e^{x}+x^{2}\right)$
$\Rightarrow e^{y}dy=\left(e^{x}+x^{2}\right)dx$
On integrating, we get
$e^{y}=e^{x}+\frac{x^{3}}{3}+c$
$\Rightarrow e^{y}-e^{x}=\frac{x^{3}}{3}+c$