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: Concept
Use exponent rules to separate the variables.

Step 2: Meaning
$\frac{dy}{dx} = e^{x}e^{-y} + x^{2}e^{-y} = e^{-y}(e^{x} + x^{2})$.

Step 3: Analysis
Separating variables: $\frac{dy}{e^{-y}} = (e^{x} + x^{2}) dx \implies e^{y} dy = (e^{x} + x^{2}) dx$.

Step 4: Conclusion
Integrate both sides: $e^{y} = e^{x} + \frac{x^{3}}{3} + c$.
Final Answer: (B)