1. **Problem statement:** Find the general solution of the differential equation $$\frac{dy}{dx} = e^{x - y}$$ by separating variables. 2. **Step 1: Write the equation and separate variables.** $$\frac{dy}{dx} = e^{x - y} = e^x \cdot e^{-y}$$ Rewrite as: $$e^y dy = e^x dx$$ 3. **Step 2: Integrate both sides.** $$\int e^y dy = \int e^x dx$$ 4. **Step 3: Compute the integrals.** $$\int e^y dy = e^y + C_1$$ $$\int e^x dx = e^x + C_2$$ 5. **Step 4: Combine constants and write the implicit solution.** $$e^y = e^x + C$$ where $$C = C_2 - C_1$$ is an arbitrary constant. 6. **Step 5: Solve for $$y$$ explicitly.** Take the natural logarithm on both sides: $$y = \ln(e^x + C)$$ **Final answer:** $$y = \ln(e^x + C)$$ This matches the given solution.