What is the solution for 9y^{''}+6y^'+y=x^2 ?

The solution for 9y^{''}+6y^'+y=x^2 is y=c_{1}e^{-x/3}+c_{2}e^{-x/3}x+x^2-12x+54

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9y^{''}+6y^'+y=x^2

9 y^{^{''}} + 6 y^{^{'}} + y = x^{2}

y = c_{1} e^{- \frac{x}{3}} + c_{2} e^{- \frac{x}{3}} x + x^{2} - 12 x + 54

Solution steps

9 y^{''} (x) + 6 y^{'} (x) + y = x^{2}

Solve linear ODE:

y = c_{1} e^{- \frac{x}{3}} + c_{2} e^{- \frac{x}{3}} x + x^{2} - 12 x + 54

y = c_{1} e^{- \frac{x}{3}} + c_{2} e^{- \frac{x}{3}} x + x^{2} - 12 x + 54

y^{^{''}} + 10 y^{^{'}} + 26 y = 2 e^{- 5 x} cos (x)

(D^{2} + 3 D + 2) y = 40 e^{3 x}

y^{^{''}} - y^{^{'}} - 6 y = sin (3 x)

y^{^{'''}} - y^{^{'}} = 2 cos (x)

y^{^{'''}} - 5 y^{^{''}} + 8 y^{^{'}} - 4 y = e^{2 x} - 4 x + 4

What is the solution for 9y^{''}+6y^'+y=x^2 ?
The solution for 9y^{''}+6y^'+y=x^2 is y=c_{1}e^{-x/3}+c_{2}e^{-x/3}x+x^2-12x+54