Solution of differential equation \( \displaystyle\frac{dy}{dx}=e^{x-y}+3x^2e^{-y} \) is:
(where C is an arbitrary constant)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( e^y-e^x=x^3+\mathrm{C} \)
Taking \( e^{-y} \) common, the variables separate:
\( \displaystyle \frac{dy}{dx}=e^{-y}(e^x+3x^2) \)
\( \displaystyle e^y\,dy=(e^x+3x^2)\,dx \)
Integrating both sides,
\( \displaystyle e^y=e^x+x^3+C \)
\( \displaystyle e^y-e^x=x^3+C \)