What is the solution for y^{''}+6y^'+9y=6xe^{-3x}+27x+27 ?

The solution for y^{''}+6y^'+9y=6xe^{-3x}+27x+27 is y=c_{1}e^{-3x}+c_{2}xe^{-3x}+e^{-3x}x^3+3x+1

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y^{''}+6y^'+9y=6xe^{-3x}+27x+27

y^{^{''}} + 6 y^{^{'}} + 9 y = 6 x e^{- 3 x} + 27 x + 27

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + e^{- 3 x} x^{3} + 3 x + 1

Solution steps

y^{''} (x) + 6 y^{'} (x) + 9 y = 6 x e^{- 3 x} + 27 x + 27

Solve linear ODE:

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + e^{- 3 x} x^{3} + 3 x + 1

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + e^{- 3 x} x^{3} + 3 x + 1

y^{^{''}} + 3 y^{^{'}} - 10 y = - 42 t + 20 t^{2}

y^{^{''}} - 4 y^{^{'}} + 4 y = 2 e^{5 t}

4 q^{^{''}} + 8 q^{^{'}} + 80 q = 88

y^{^{''}} + y = sin (e^{- x}) (e^{- 2 x})

\frac{d ^{2} y}{d x ^{2}} - 25 y = 3 x^{2} - 2 + e^{x}

What is the solution for y^{''}+6y^'+9y=6xe^{-3x}+27x+27 ?
The solution for y^{''}+6y^'+9y=6xe^{-3x}+27x+27 is y=c_{1}e^{-3x}+c_{2}xe^{-3x}+e^{-3x}x^3+3x+1