1. The problem is to identify two correct statements about the differential equation $$\frac{dX}{dt} + e^{2t} \cdot X = t^2$$. 2. First, identify the dependent and independent variables. In this equation, $X$ depends on $t$, so $X$ is the dependent variable and $t$ is the independent variable. This confirms statement A is correct and B is incorrect. 3. To solve the linear first-order differential equation, use the integrating factor method. The standard form is: $$\frac{dX}{dt} + P(t)X = Q(t)$$ where here $P(t) = e^{2t}$ and $Q(t) = t^2$. 4. The integrating factor $\mu(t)$ is: $$\mu(t) = e^{\int P(t) dt} = e^{\int e^{2t} dt}$$ Calculate the integral: $$\int e^{2t} dt = \frac{1}{2} e^{2t} + C$$ Ignoring the constant, the integrating factor is: $$\mu(t) = e^{\frac{1}{2} e^{2t}}$$ 5. Multiply both sides of the differential equation by $\mu(t)$: $$e^{\frac{1}{2} e^{2t}} \frac{dX}{dt} + e^{\frac{1}{2} e^{2t}} e^{2t} X = t^2 e^{\frac{1}{2} e^{2t}}$$ 6. The left side is the derivative of $X \cdot \mu(t)$: $$\frac{d}{dt} \left(X e^{\frac{1}{2} e^{2t}}\right) = t^2 e^{\frac{1}{2} e^{2t}}$$ 7. Integrate both sides: $$X e^{\frac{1}{2} e^{2t}} = \int t^2 e^{\frac{1}{2} e^{2t}} dt + C$$ 8. Solve for $X(t)$: $$X(t) = e^{-\frac{1}{2} e^{2t}} \int t^2 e^{\frac{1}{2} e^{2t}} dt + Ce^{-\frac{1}{2} e^{2t}}$$ 9. Comparing with the options, statement D matches this solution form exactly. 10. Therefore, the two correct statements are A and D. Final answer: A and D are correct.