General solution of the differential equation
\( \displaystyle\frac{dy}{dx}=e^{x-y}+3x^2e^{-y} \) is: (Where \( c \) is an arbitrary constant)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( e^y=e^x+x^3+c \)
Since both terms contain \( e^{-y} \), 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:
\( \displaystyle e^y=e^x+x^3+c \)