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The full list of solutions to the equation $y''-2y'-3y=2xe^{-x}$ is given by:

A

$C_1e^{3x}+C_2e^{-x}-\frac{1}{4}x^2e^{-x}-\frac{1}{8}xe^{-x}$

B

$C_1e^{3x}+C_2e^{-x}-\frac{1}{2}xe^{-x}$

C

$C_1e^{3x}+C_2e^{-x}-\frac{1}{2}x^2e^{-x}-\frac{1}{4}xe^{-x}$

D

$C_1e^{3x}+C_2e^{-x}-\frac{1}{4}x^2e^{-x}$

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