Question

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

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

Answer 1

Answer: (C)

$e^{y}=2e^{x}+\frac{x^{3}}{3}+c$

Answer 2

Given, $\frac{d y}{d x}=2 e^{x-y}+ x^{2} e^{-y}$
$\Rightarrow \frac{d y}{d x}=2 e^{x} \cdot \frac{1}{e^{y}}+x^{2} \cdot \frac{1}{e^{y}}$
$\Rightarrow \frac{d y}{d x}=\frac{1}{e^{y}}\left(2 e^{x}+x^{2}\right)$
$\Rightarrow e^{y} d y=\left(2 e^{x}+x^{2}\right) d x$
On Integrating both sides, we get
$\int e^{y} d y=\int\left(2 e^{x}+x^{2}\right) d x$
$\Rightarrow e^{y}=2 e^{x}+\frac{x^{3}}{3}+C$
which is the required solution.