Step-by-step Solution:

\[ \frac{dy}{dx} = e^{x - y} + x^2 e^{-y} \] \[ \Rightarrow dy = \left( e^{x - y} + x^2 e^{-y} \right) dx \] \[ \Rightarrow dy = \frac{e^x + x^2}{e^y} dx \] \[ \Rightarrow e^y dy = \left( e^x + x^2 \right) dx \] \[ \Rightarrow \int e^y dy = \int (e^x + x^2) dx \] \[ \Rightarrow e^y = e^x + \frac{x^3}{3} + C \] Thus, the required solution is: \[ e^y = e^x + \frac{x^3}{3} + C \]