What is the solution for (D^2+3D+2)y=1+3x+x^2 ?

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

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(D^2+3D+2)y=1+3x+x^2

(D^{2} + 3 D + 2) y = 1 + 3 x + x^{2}

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

Solution steps

(D^{2} + 3 D + 2) y = 1 + 3 x + x^{2}

Solve linear ODE:

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

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

y^{^{''}} - 4 y = cos (x) - sin (x)

y^{^{''}} + 8 y^{^{'}} + 12 y = x e^{2 x}

x^{^{''}} + 2 x^{^{'}} + 1 x = e^{- 2 t}

y^{^{''}} + 9 y = 4 sin (3 x)

y^{^{''}} + 4 y^{^{'}} + 4 y = x^{2} \cdot e^{- 2 x} + 1

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