What is the solution for y^{'''}+2y^{''}+y^'=2x^2 ?

The solution for y^{'''}+2y^{''}+y^'=2x^2 is y=c_{1}+c_{2}e^{-x}+c_{3}xe^{-x}+(2x^3)/3-4x^2+12x

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y^{'''}+2y^{''}+y^'=2x^2

y^{^{'''}} + 2 y^{^{''}} + y^{^{'}} = 2 x^{2}

y = c_{1} + c_{2} e^{- x} + c_{3} x e^{- x} + \frac{2 x ^{3}}{3} - 4 x^{2} + 12 x

Solution steps

y^{'''} (x) + 2 y^{''} (x) + y^{'} (x) = 2 x^{2}

Solve linear ODE:

y = c_{1} + c_{2} e^{- x} + c_{3} x e^{- x} + \frac{2 x ^{3}}{3} - 4 x^{2} + 12 x

y = c_{1} + c_{2} e^{- x} + c_{3} x e^{- x} + \frac{2 x ^{3}}{3} - 4 x^{2} + 12 x

(d^{2} + 4 d + 4) y = 10 e^{- x} cos (x)

(d^{2} + 5 d + 6) \cdot y = e^{x} + e^{- x}

y^{^{'''}} - y^{^{''}} + y^{^{'}} - y = x^{2} + x

(d^{3} - 3 d + 2) \cdot y = x + 1

(d^{2} + 4) \cdot y = 4 tan^{2} (x)

What is the solution for y^{'''}+2y^{''}+y^'=2x^2 ?
The solution for y^{'''}+2y^{''}+y^'=2x^2 is y=c_{1}+c_{2}e^{-x}+c_{3}xe^{-x}+(2x^3)/3-4x^2+12x