The general solution of the differential equation \(\displaystyle\frac{dy}{dx} = xe^{-y}\) is (where C is an arbitrary constant)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(e^{y} = \displaystyle\frac{x^2}{2} + C\)
Since \(e^{-y} = \frac{1}{e^{y}}\), the variables can be separated by multiplying by \(e^{y}\):
\(e^{y}\,dy = x\,dx\)
Integrating both sides:
\( \displaystyle \int e^{y}\,dy = \int x\,dx \)
\( \displaystyle e^{y} = \frac{x^2}{2} + C \)