Question

General solution of the differential equation \[ \frac{dy}{dx}=e^{x-y}+3x^{2}e^{-y} \] is:

• A. \(e^{-y}+e^{x}=x^{3}+c \)
• B. \(e^{-y}=e^{x}+x^{3}+c \)
• C. \(e^{y}=e^{x}-x^{3}+c \)
• D. \(e^{y}=e^{x}+x^{3}+c \)

Hint:

Whenever you see $e^{x-y}$ or $e^{-y}$ together, try factoring $e^{-y}$ first to separate variables easily.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Rewrite the equation to make it separable or linear. Given \( \frac{dy}{dx} = e^x - y + 3x^2 e^{-y} \) seems to have a typo. If it were \( e^y \frac{dy}{dx} = e^x + 3x^2 \):

Step 2: Detailed Explanation:
Multiply by \( e^y \): \( e^y \frac{dy}{dx} = e^x + 3x^2 \).
Now integrate both sides with respect to \( x \):
\( \int e^y dy = \int (e^x + 3x^2) dx \)
\( e^y = e^x + x^3 + c \).

Step 3: Final Answer:
The general solution is \( e^y = e^x + x^3 + c \), which is option (D).