The general solution of the differential equation \(\displaystyle\frac{dy}{dx} = e^{x + y} + x^2e^{y}\) is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(e^{x} + e^{-y} + \displaystyle\frac{x^3}{3} = C\)
Since \(e^{x + y} = e^{x}e^{y}\), the factor \(e^{y}\) is common to both terms:
\( \displaystyle \frac{dy}{dx} = e^{y}\left(e^{x} + x^2\right) \)
So the variables separate:
\(e^{-y}\,dy = \left(e^{x} + x^2\right)dx\)
Integrating both sides:
\( \displaystyle -e^{-y} = e^{x} + \frac{x^3}{3} + C_1 \)
Moving \(e^{-y}\) to the right and writing \(C = -C_1\):
\( \displaystyle e^{x} + e^{-y} + \frac{x^3}{3} = C \)