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

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

The correct answer is Option (2) → $e^y - e^x = \frac{x^3}{3} + C$ ##

Given that, $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$

$\Rightarrow \frac{dy}{dx} = e^x e^{-y} + x^2 e^{-y}$

$\Rightarrow \frac{dy}{dx} = \frac{e^x + x^2}{e^y}$

$\Rightarrow e^y \, dy = (e^x + x^2) \, dx \quad \text{[using variable separable method]}$

On integrating both sides, we get

$\int e^y \, dy = \int (e^x + x^2) \, dx$

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

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