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

Step 1: Understanding the Concept:
This is a variable separable differential equation. We can split the exponential term using the property $e^{a+b} = e^a \cdot e^b$.
Step 2: Formula Application:
$$\frac{dy}{dx} = e^x \cdot e^y$$ Rearrange to bring $y$ terms with $dy$ and $x$ terms with $dx$: $$\frac{dy}{e^y} = e^x dx \implies e^{-y} dy = e^x dx$$
Step 3: Explanation:
Integrate both sides: $$\int e^{-y} dy = \int e^x dx$$ $$-e^{-y} = e^x + C'$$ $$e^x + e^{-y} = -C'$$ Let $-C' = C$. Then $e^x + e^{-y} = C$.
Step 4: Final Answer:
The correct option is (a).