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
$e^y=e^x+\frac{x^3}{3}+c$
$\Rightarrow e^y-e^x=\frac{x^3}{3}+c$