The general solution of differential equation \(\displaystyle\frac{dy}{dx}= e^{x+y}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(e^x + e^{-y}= C\), where C is an arbitrary constant
Since \(e^{x+y} = e^x e^y\), the variables separate:
\( \displaystyle \frac{dy}{dx} = e^x e^y \)
\(e^{-y}\,dy = e^x\,dx\)
Integrating both sides:
\( \displaystyle \int e^{-y}\,dy = \int e^x\,dx \)
\(-e^{-y} = e^x + c\)
\(e^x + e^{-y} = -c = C\)
Why the other options are incorrect
Option (1) \(e^x + e^y= C\), where C is an arbitrary constant — Incorrect. Differentiating, \(e^x + e^y\frac{dy}{dx} = 0\), so \(\frac{dy}{dx} = -e^{x-y}\), not \(e^{x+y}\).
Option (3) \(e^x - e^{-y}= C\), where C is an arbitrary constant — Incorrect. This takes \(\int e^{-y}\,dy = e^{-y}\). The minus sign from \(\int e^{-y}\,dy = -e^{-y}\) is lost.
Option (4) \(e^x - e^y= C\), where C is an arbitrary constant — Incorrect. Differentiating, \(e^x - e^y\frac{dy}{dx} = 0\), so \(\frac{dy}{dx} = e^{x-y}\), not \(e^{x+y}\).
NCERT (2026-27 edition):
Differential Equations with Variables Separable
