1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = 2y$$ and find the general solution. 2. **Use the method of separation of variables:** Rewrite the equation as $$\frac{dy}{y} = 2 \, dx$$. 3. **Integrate both sides:** $$\int \frac{1}{y} \, dy = \int 2 \, dx$$ which gives $$\ln|y| = 2x + C$$ where $C$ is the constant of integration. 4. **Solve for $y$:** Exponentiate both sides: $$y = e^{2x + C} = e^C e^{2x}$$ Let $C_1 = e^C$, so $$y = C_1 e^{2x}$$ This is the general solution. 5. **Find the particular solution given $y(2) = 4$:** Substitute $x=2$ and $y=4$: $$4 = C_1 e^{2 \cdot 2} = C_1 e^4$$ Solve for $C_1$: $$C_1 = \frac{4}{e^4} = 4 e^{-4}$$ 6. **Write the particular solution:** $$y = 4 e^{-4} e^{2x} = 4 e^{2x - 4}$$ **Note:** The expression $y = \sqrt{2x^2 + 4}$ is unrelated to the solution of this differential equation.