What is the solution for (D^3-3D-2)y=x^2 ?

The solution for (D^3-3D-2)y=x^2 is y=c_{1}e^{2x}+c_{2}e^{-x}+c_{3}xe^{-x}-(x^2)/2+(3x)/2-9/4

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(D^3-3D-2)y=x^2

(D^{3} - 3 D - 2) y = x^{2}

y = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} x e^{- x} - \frac{x ^{2}}{2} + \frac{3 x}{2} - \frac{9}{4}

Solution steps

(D^{3} - 3 D - 2) y = x^{2}

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} x e^{- x} - \frac{x ^{2}}{2} + \frac{3 x}{2} - \frac{9}{4}

y = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} x e^{- x} - \frac{x ^{2}}{2} + \frac{3 x}{2} - \frac{9}{4}

What is the solution for (D^3-3D-2)y=x^2 ?
The solution for (D^3-3D-2)y=x^2 is y=c_{1}e^{2x}+c_{2}e^{-x}+c_{3}xe^{-x}-(x^2)/2+(3x)/2-9/4