Question

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^{2}e^{-y}$ is

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

Hint:

When you see $e^{x-y}$, immediately think $e^x/e^y$ to separate your $x$'s and $y$'s.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We simplify the right side of the equation using exponent rules to separate the \(x\) and \(y\) variables.
Step 2: Key Formula or Approach:
1. Use \( e^{x-y} = e^x \cdot e^{-y} \).
2. Factor out \( e^{-y} \) and separate variables.
Step 3: Detailed Explanation:
\[ \frac{dy}{dx} = e^x e^{-y} + x^2 e^{-y} \] \[ \frac{dy}{dx} = e^{-y} (e^x + x^2) \] Separate the variables: \[ \frac{dy}{e^{-y}} = (e^x + x^2) dx \] \[ e^y dy = (e^x + x^2) dx \] Integrate both sides: \[ \int e^y dy = \int (e^x + x^2) dx \] \[ e^y = e^x + \frac{x^3}{3} + c \]
Step 4: Final Answer:
The solution is \( e^y = e^x + \frac{x^3}{3} + c \).