1. The problem is to eliminate the constants $C_1$, $C_2$, and $C_3$ from the function $$y = C_1 e^{2x} + C_2 x e^{2x} + C_3 x^2 e^{2x}$$ and find the order of the resulting differential equation. 2. Since there are three arbitrary constants, the order of the differential equation formed by eliminating these constants will be equal to the number of constants, which is 3. 3. To confirm, differentiate $y$ three times to get three equations involving $y$, $y'$, $y''$, and $y'''$: $$y = C_1 e^{2x} + C_2 x e^{2x} + C_3 x^2 e^{2x}$$ $$y' = 2 C_1 e^{2x} + C_2 (e^{2x} + 2x e^{2x}) + C_3 (2x e^{2x} + 2x^2 e^{2x})$$ $$y'' = 4 C_1 e^{2x} + C_2 (4 e^{2x} + 4x e^{2x}) + C_3 (4x e^{2x} + 4x^2 e^{2x})$$ $$y''' = 8 C_1 e^{2x} + C_2 (12 e^{2x} + 6x e^{2x}) + C_3 (12x e^{2x} + 6x^2 e^{2x})$$ 4. Using these, you can eliminate $C_1$, $C_2$, and $C_3$ to get a third-order differential equation. 5. Therefore, the order of the resulting differential equation is 3. Answer: B 3