Question

The general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \) is _____

• A. \( e^{x} + e^{-y} = C \)
• B. \( e^{-x} + e^y = C \)
• C. \( e^x + e^y = C \)
• D. \( e^{-x} + e^{-y} = C \)

Hint:

Always integrate RHS directly when equation is in the form \( \frac{dy}{dx} = f(x) \).

Answer 1

The Correct Option is A

Concept: To solve: \[ \frac{dy}{dx} = f(x) \Rightarrow y = \int f(x)\,dx \]
Step 1: Integrate both sides. \[ y = \int (e^x + e^{-y}) dx \]
Step 2: \[ y = e^x - e^{-y} + C \]
Step 3: Rearranging, \[ e^x + e^{-y} = C \]