1. **State the problem:** Solve the first-order linear differential equation $$y' + e^{2x} y = e^x.$$\n\n2. **Identify the type and formula:** This is a linear ODE of the form $$y' + P(x)y = Q(x)$$ where $$P(x) = e^{2x}$$ and $$Q(x) = e^x.$$\n\n3. **Find the integrating factor (IF):** $$\mu(x) = e^{\int P(x) dx} = e^{\int e^{2x} dx}.$$\nCalculate the integral: $$\int e^{2x} dx = \frac{1}{2} e^{2x} + C.$$\nIgnoring the constant, $$\mu(x) = e^{\frac{1}{2} e^{2x}}.$$\n\n4. **Multiply the entire ODE by the integrating factor:**\n$$e^{\frac{1}{2} e^{2x}} y' + e^{2x} e^{\frac{1}{2} e^{2x}} y = e^x e^{\frac{1}{2} e^{2x}}.$$\n\n5. **Recognize the left side as a derivative:**\n$$\frac{d}{dx} \left(y e^{\frac{1}{2} e^{2x}}\right) = e^x e^{\frac{1}{2} e^{2x}}.$$\n\n6. **Integrate both sides:**\n$$y e^{\frac{1}{2} e^{2x}} = \int e^x e^{\frac{1}{2} e^{2x}} dx + C.$$\n\n7. **Final solution:** The integral on the right side does not simplify to elementary functions, so the implicit solution is\n$$y = e^{-\frac{1}{2} e^{2x}} \left( \int e^x e^{\frac{1}{2} e^{2x}} dx + C \right).$$\n\nThis is the general solution to the differential equation.