Step 1: Understanding the Concept:
A separable differential equation is an equation where the variables can be rearranged so that all terms containing \( y \) appear on one side and all terms containing \( x \) appear on the other side.
The general form is \( h(y) dy = g(x) dx \).
Once separated, the solution can be found by integrating both sides independently.
In this specific problem, we are given an equation where \( x \) and \( y \) are combined in exponential powers.
The key to separation here is applying the algebraic rules of exponents to "break apart" these terms.
Step 2: Key Formula or Approach:
Use the law of exponents: \( e^{a-b} = e^a \cdot e^{-b} \).
Separate the variables into the form \( \int h(y) dy = \int g(x) dx \).
Integrate using basic rules: \( \int e^u du = e^u \) and \( \int x^n dx = \frac{x^{n+1}}{n+1} \).
Step 3: Detailed Explanation:
The given equation is:
\[ \frac{dy}{dx} = e^{x-y} + x^2 e^{-y} \]
First, rewrite the first term on the right using the law of exponents:
\[ \frac{dy}{dx} = e^x \cdot e^{-y} + x^2 \cdot e^{-y} \]
Observe that both terms on the right-hand side share a common factor of \( e^{-y} \).
Factoring this out gives us:
\[ \frac{dy}{dx} = e^{-y} (e^x + x^2) \]
Now, we separate the variables by multiplying both sides by \( dx \) and dividing by \( e^{-y} \):
\[ \frac{1}{e^{-y}} dy = (e^x + x^2) dx \]
Since \( \frac{1}{e^{-y}} = e^y \), the equation becomes:
\[ e^y dy = (e^x + x^2) dx \]
Now that the variables are fully separated, we integrate both sides:
\[ \int e^y dy = \int (e^x + x^2) dx \]
Integrating the left side:
\[ \int e^y dy = e^y \]
Integrating the right side term-by-term:
\[ \int e^x dx = e^x \]
\[ \int x^2 dx = \frac{x^3}{3} \]
Combining everything and adding the constant of integration \( C \):
\[ e^y = e^x + \frac{x^3}{3} + C \]
Step 4: Final Answer:
The general solution of the differential equation is \( e^y = e^x + \frac{x^3}{3} + C \).
This matches Option (A).