1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = e^{x+y}$$. 2. **Rewrite the equation:** Use the property of exponents to separate variables: $$\frac{dy}{dx} = e^x \cdot e^y$$. 3. **Separate variables:** Rearrange to isolate $y$ terms on one side and $x$ terms on the other: $$\frac{dy}{e^y} = e^x dx$$. 4. **Integrate both sides:** $$\int e^{-y} dy = \int e^x dx$$. 5. **Compute the integrals:** $$-e^{-y} = e^x + C$$ where $C$ is the constant of integration. 6. **Solve for $y$:** Multiply both sides by $-1$: $$e^{-y} = -e^x - C$$. Rewrite $-C$ as a new constant $C_1$: $$e^{-y} = -e^x + C_1$$. Take the natural logarithm: $$-y = \ln\left(-e^x + C_1\right)$$. Multiply both sides by $-1$: $$y = -\ln\left(C_1 - e^x\right)$$. 7. **Final solution:** $$y = -\ln\left(C - e^x\right)$$ where $C$ is an arbitrary constant. --- **Summary:** We separated variables, integrated both sides, and solved for $y$ to find the implicit general solution.