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