The general solution of the differential equation \(\displaystyle\frac{dy}{dx}=e^{x-y}+x^2e^{-y}\) is equal to:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(e^y=e^x+\displaystyle\frac{x^3}{3}+c\), where c is an arbitrary constant.
Both terms on the right contain \(e^{-y}\), so the variables separate:
\( \displaystyle \frac{dy}{dx} = e^{-y}\left(e^{x} + x^2\right) \)
\(e^{y}\,dy = \left(e^{x} + x^2\right)dx\)
Integrating both sides:
\( \displaystyle \int e^{y}\,dy = \int \left(e^{x} + x^2\right)dx \)
\( \displaystyle e^{y} = e^{x} + \frac{x^3}{3} + c \)