\[\large y''+6y'+9y=x^2+xe^{-3x}+\cos{x}\]

Which of the options below is the solution \(y=y(x)\) of the differential equation above?

\(\begin{align} \quad \text{A:} \quad & c_{1}e^{-3x}+c_{2}xe^{-3x}+\dfrac{x^2}{9}+\dfrac{3\sin{x}}{50}+\dfrac{2\cos{x}}{25} \\ \text{B:} \quad & c_{1}e^{-3x} +c_{2}x^3e^{-3x} +\dfrac{x^2}{9}-\dfrac{4x}{27}+\dfrac{3\sin{x}}{50} \\ \text{C:} \quad & c_{1}e^{-3x} + c_{2}xe^{-3x} +\dfrac{1}{6}x^2e^{-3x}+\dfrac{x^2}{9}-\frac{4x}{27}+\dfrac{2}{27} +\dfrac{3\sin{x}}{50} -\dfrac{2\cos{x}}{25} \\ \text{D:} \quad & c_{1}e^{-3x}+c_{2}xe^{-3x} + \dfrac{1}{6}x^3e^{-3x} +\dfrac{x^2}{9}-\dfrac{4x}{27}+\dfrac{2}{27}+\dfrac{3\sin{x}}{50}+\dfrac{2\cos{x}}{25} \end{align} \)

(This problem is part of the set Extraordinary Differential Equations.)

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